{"id":296,"date":"2022-01-03T10:38:40","date_gmt":"2022-01-03T02:38:40","guid":{"rendered":"https:\/\/www.lazybirds.top\/?p=296"},"modified":"2022-07-09T11:08:59","modified_gmt":"2022-07-09T03:08:59","slug":"%e7%bb%84%e5%90%88%e6%95%b0%e5%ad%a6-%e6%9c%9f%e6%9c%ab%e5%a4%8d%e4%b9%a0","status":"publish","type":"post","link":"https:\/\/www.lazybirds.top\/?p=296","title":{"rendered":"\u7ec4\u5408\u6570\u5b66 &#8211; \u671f\u672b\u590d\u4e60"},"content":{"rendered":"\n<!--more-->\n\n\n\n<meta charset=\"utf-8\">\n      <meta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\">\n      \n      <link rel=\"stylesheet\" href=\"https:\/\/www.lazybirds.top\/katex\/katex.min.css\">\n      \n      \n      \n      \n      \n      \n      \n      \n      \n      <style>\n      \/**\n * prism.js Github theme based on GitHub's theme.\n * @author Sam Clarke\n *\/\ncode[class*=\"language-\"],\npre[class*=\"language-\"] {\n  color: #333;\n  background: none;\n  font-family: Consolas, 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stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mo>\u220f<\/mo><mrow><mi>o<\/mi><mi>d<\/mi><mi>d<\/mi><mtext>&nbsp;<\/mtext><mi>k<\/mi><mo>\u2265<\/mo><mn>1<\/mn><\/mrow><\/munder><mfrac><mn>1<\/mn><mrow><mn>1<\/mn><mo>\u2212<\/mo><msup><mi>x<\/mi><mi>k<\/mi><\/msup><\/mrow><\/mfrac><mo>=<\/mo><munder><mo>\u220f<\/mo><mrow><mi>k<\/mi><mo>\u2265<\/mo><mn>1<\/mn><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><msup><mi>x<\/mi><mrow><mn>2<\/mn><mi>k<\/mi><\/mrow><\/msup><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><msup><mi>x<\/mi><mi>k<\/mi><\/msup><\/mrow><\/mfrac><mo>=<\/mo><munder><mo>\u220f<\/mo><mrow><mi>k<\/mi><mo>\u2265<\/mo><mn>1<\/mn><\/mrow><\/munder><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mi>x<\/mi><mi>k<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>P<\/mi><mrow><mi>d<\/mi><mi>i<\/mi><mi>f<\/mi><mi>f<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u223c<\/mo><msup><mi>e<\/mi><mrow><mi>C<\/mi><msqrt><mi>n<\/mi><\/msqrt><\/mrow><\/msup><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{gather*}\n    P(n):=\\#\\{(x_1,x_2,\\dots,x_{m})|n=x_1+x_2+\\cdots+x_m,x_1\\ge x_2\\ge\\cdots\\ge x_m\\ge 1,m\\in\\mathbb{N}\\}\\\\\n    P(n,m):=\\#\\{(x_1,x_2,\\dots,x_{m})|n=x_1+x_2+\\cdots+x_m,x_1\\ge x_2\\ge\\cdots\\ge x_m\\ge 1\\}\\\\\n    P(n)=P(n-1)+\\sum_{m=1}^{\\lfloor\\frac{n}{2}\\rfloor}P(n-m,m)\\\\\n    P(x):=\\sum_{n\\ge 0}P(n)x^n=\\prod_{k\\ge 1}\\frac{1}{1-x^k}\\\\\n    P_{odd}(n)=\\prod_{odd\\ k\\ge1}\\frac{1}{1-x^k}=\\prod_{k\\ge 1}\\frac{1-x^{2k}}{1-x^k}=\\prod_{k\\ge1}(1+x^k)=P_{diff}(n)\\\\\n    P(n)\\sim e^{C\\sqrt{n}}\n\\end{gather*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:14.473183500000001em;vertical-align:-6.9865917500000005em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:7.4865917500000005em;\"><span style=\"top:-10.67339675em;\"><span class=\"pstrut\" style=\"height:4.0268049999999995em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">#<\/span><span class=\"mopen\">{(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord mathbb\">N<\/span><span class=\"mclose\">}<\/span><\/span><\/span><span style=\"top:-9.17339675em;\"><span class=\"pstrut\" style=\"height:4.0268049999999995em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">#<\/span><span class=\"mopen\">{(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">}<\/span><\/span><\/span><span style=\"top:-6.48659175em;\"><span class=\"pstrut\" style=\"height:4.0268049999999995em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.026805em;\"><span style=\"top:-1.882887em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.050005em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span style=\"top:-4.451805em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">\u230a<\/span><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6915428571428572em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><span class=\"mclose mtight\">\u230b<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.267113em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span style=\"top:-3.5980387499999997em;\"><span class=\"pstrut\" style=\"height:4.0268049999999995em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8828869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mrel mtight\">\u2265<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.362292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7143919999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mrel mtight\">\u2265<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.397292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7751079999999999em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><span style=\"top:-0.374638749999999em;\"><span class=\"pstrut\" style=\"height:4.0268049999999995em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">o<\/span><span class=\"mord mathnormal mtight\">dd<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">o<\/span><span class=\"mord mathnormal mtight\">dd<\/span><span class=\"mspace mtight\"><span class=\"mtight\">&nbsp;<\/span><\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mrel mtight\">\u2265<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.397292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7751079999999999em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mrel mtight\">\u2265<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.397292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.526108em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7751079999999999em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.849108em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mrel mtight\">\u2265<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.397292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361079999999999em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">d<\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10764em;\">ff<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span style=\"top:2.299786750000001em;\"><span class=\"pstrut\" style=\"height:4.0268049999999995em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u223c<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9771335000000001em;\"><span style=\"top:-3.1130000000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"mord sqrt mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8059050000000001em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord mtight\" style=\"padding-left:0.833em;\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><span style=\"top:-2.765905em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail mtight\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.234095em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:6.9865917500000005em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E7%B4%A0%E6%95%B0%E5%88%86%E5%B8%83\">\u7d20\u6570\u5206\u5e03<\/h2>\n\n<h3 class=\"mume-header\" id=\"bertrand-chebyshev-%E5%AE%9A%E7%90%86\">Bertrand-Chebyshev \u5b9a\u7406<\/h3>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mn>2<\/mn><mi>n<\/mi><mo stretchy=\"false\">]<\/mo><mtext>\u95f4\u5b58\u5728\u7d20\u6570<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">[n,2n]\u95f4\u5b58\u5728\u7d20\u6570<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">]<\/span><span class=\"mord cjk_fallback\">\u95f4\u5b58\u5728\u7d20\u6570<\/span><\/span><\/span><\/span><\/span><br>\n\u8bc1\u660e\u601d\u8def\uff1a<br>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"center\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>\u03b1<\/mi><mi>p<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>:<\/mo><mo>=<\/mo><mo stretchy=\"false\">\u230a<\/mo><mfrac><mi>n<\/mi><mi>p<\/mi><\/mfrac><mo stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mo stretchy=\"false\">\u230a<\/mo><mfrac><mi>n<\/mi><msup><mi>p<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mo stretchy=\"false\">\u230a<\/mo><mfrac><mi>n<\/mi><msup><mi>p<\/mi><mn>3<\/mn><\/msup><\/mfrac><mo stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mo>\u22ef<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">!<\/mo><\/mrow><mrow><mi>n<\/mi><mo stretchy=\"false\">!<\/mo><mi>n<\/mi><mo stretchy=\"false\">!<\/mo><\/mrow><\/mfrac><mo>=<\/mo><munder><mo>\u220f<\/mo><mrow><mi>p<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>p<\/mi><mo>\u2264<\/mo><mn>2<\/mn><mi>n<\/mi><\/mrow><\/munder><msup><mi>p<\/mi><mrow><msub><mi>\u03b1<\/mi><mi>p<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><msub><mi>\u03b1<\/mi><mi>p<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msup><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><munder><mo>\u220f<\/mo><mrow><mi>p<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>p<\/mi><mo>\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder><msup><mi>p<\/mi><mrow><msub><mi>\u03b1<\/mi><mi>p<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><msub><mi>\u03b1<\/mi><mi>p<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msup><mo>\u2264<\/mo><msup><mn>2<\/mn><mrow><msqrt><mn>2<\/mn><\/msqrt><msub><mrow><mi>log<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msub><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mrow><\/msup><mo>\u22c5<\/mo><msup><mn>2<\/mn><mrow><mfrac><mn>4<\/mn><mn>3<\/mn><\/mfrac><mi>n<\/mi><\/mrow><\/msup><mo>\u22c5<\/mo><mn>1<\/mn><mo>\u21d2<\/mo><munder><mo>\u220f<\/mo><mrow><mi>p<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>p<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mn>2<\/mn><mi>n<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/munder><msup><mi>p<\/mi><mrow><msub><mi>\u03b1<\/mi><mi>p<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><msub><mi>\u03b1<\/mi><mi>p<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msup><mo>&gt;<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{gather*}\n    \\alpha_p(n):=\\lfloor\\frac{n}{p}\\rfloor+\\lfloor\\frac{n}{p^2}\\rfloor+\\lfloor\\frac{n}{p^3}\\rfloor+\\cdots\\\\\n    \\binom{2n}{n}=\\frac{(2n)!}{n!n!}=\\prod_{prime\\ p\\le 2n}p^{\\alpha_p(2n)-2\\alpha_p(n)}\\\\\n    \\prod_{prime\\ p\\le n}p^{\\alpha_p(2n)-2\\alpha_p(n)}\\le 2^{\\sqrt{2}\\log_2{2n}}\\cdot2^{\\frac{4}{3}n}\\cdot 1\\Rightarrow\\prod_{prime\\ p\\in(n,2n]}p^{\\alpha_p(2n)-2\\alpha_p(n)}&gt;1\n\\end{gather*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:8.3197735em;vertical-align:-3.9098867500000005em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:4.40988675em;\"><span style=\"top:-6.75232675em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">\u03b1<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15139200000000003em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.0037em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">\u230a<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose\">\u230b<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mopen\">\u230a<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.740108em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose\">\u230b<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mopen\">\u230a<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.740108em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose\">\u230b<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"minner\">\u22ef<\/span><\/span><\/span><span style=\"top:-4.12188675em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.427em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">!<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">!<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)!<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8723309999999997em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">p<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r<\/span><span class=\"mord mathnormal mtight\">im<\/span><span class=\"mord mathnormal mtight\">e<\/span><span class=\"mspace mtight\"><span class=\"mtight\">&nbsp;<\/span><\/span><span class=\"mord mathnormal mtight\">p<\/span><span class=\"mrel mtight\">\u2264<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.413777em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9379999999999998em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">\u03b1<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.16454285714285716em;\"><span style=\"top:-2.357em;margin-left:-0.0037em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2818857142857143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">\u03b1<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.16454285714285716em;\"><span style=\"top:-2.357em;margin-left:-0.0037em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2818857142857143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-1.3561182499999997em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8723309999999997em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">p<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r<\/span><span class=\"mord mathnormal mtight\">im<\/span><span class=\"mord mathnormal mtight\">e<\/span><span class=\"mspace mtight\"><span class=\"mtight\">&nbsp;<\/span><\/span><span class=\"mord mathnormal mtight\">p<\/span><span class=\"mrel mtight\">\u2264<\/span><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.413777em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9379999999999998em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">\u03b1<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.16454285714285716em;\"><span style=\"top:-2.357em;margin-left:-0.0037em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2818857142857143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">\u03b1<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.16454285714285716em;\"><span style=\"top:-2.357em;margin-left:-0.0037em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2818857142857143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0519914999999997em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord sqrt mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.912845em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord mtight\" style=\"padding-left:0.833em;\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span style=\"top:-2.872845em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail mtight\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.12715500000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"><\/span><span class=\"mop mtight\"><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\" style=\"margin-right:0.01389em;\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444571428571428em;\"><span style=\"top:-2.2341314285714287em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.26586857142857145em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"><\/span><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0040200000000001em;\"><span style=\"top:-3.4130000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.808995em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">p<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r<\/span><span class=\"mord mathnormal mtight\">im<\/span><span class=\"mord mathnormal mtight\">e<\/span><span class=\"mspace mtight\"><span class=\"mtight\">&nbsp;<\/span><\/span><span class=\"mord mathnormal mtight\">p<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mclose mtight\">]<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.516005em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9379999999999998em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">\u03b1<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.16454285714285716em;\"><span style=\"top:-2.357em;margin-left:-0.0037em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2818857142857143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">\u03b1<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.16454285714285716em;\"><span style=\"top:-2.357em;margin-left:-0.0037em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2818857142857143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.9098867500000005em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 class=\"mume-header\" id=\"%E7%B4%A0%E6%95%B0%E5%AE%9A%E7%90%86\">\u7d20\u6570\u5b9a\u7406<\/h3>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c0<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u223c<\/mo><mfrac><mi>n<\/mi><mrow><msub><mrow><mi>log<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msub><mi>n<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\pi(n)\\sim\\frac{n}{\\log_2{n}}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">\u03c0<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u223c<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.0377em;vertical-align:-0.9301400000000001em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.3139999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right:0.01389em;\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.20696799999999996em;\"><span style=\"top:-2.4558600000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.24414em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9301400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 class=\"mume-header\" id=\"4k3%E5%9E%8B%E7%B4%A0%E6%95%B0%E6%9C%89%E6%97%A0%E7%A9%B7%E5%A4%9A%E4%B8%AA\">4k+3\u578b\u7d20\u6570\u6709\u65e0\u7a77\u591a\u4e2a<\/h3>\n\n<h2 class=\"mume-header\" id=\"%E4%BA%8C%E6%AC%A1%E5%89%A9%E4%BD%99\">\u4e8c\u6b21\u5269\u4f59<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"center\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>p<\/mi><mtext>&nbsp;<\/mtext><mi>p<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>a<\/mi><mi>p<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>:<\/mo><mo>=<\/mo><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.2500em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>0<\/mn><mspace width=\"1em\"><\/mspace><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mi>p<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mi>a<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>1<\/mn><mspace width=\"1em\"><\/mspace><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mi>a<\/mi><mo>\u2261<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mi>b<\/mi><mo>\u2261\u0338<\/mo><mn>0<\/mn><mspace><\/mspace><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333333333333333em\"><\/mspace><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo>\u2212<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mi>o<\/mi><mi>t<\/mi><mi>h<\/mi><mi>e<\/mi><mi>r<\/mi><mi>w<\/mi><mi>i<\/mi><mi>s<\/mi><mi>e<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{gather*}\n    p\\ prime,\\ \\left(\\frac{a}{p}\\right):=\\left\\{\\begin{aligned}0\\quad&amp; p|a\\\\1\\quad&amp; a\\equiv b^2,\\ b\\not\\equiv0\\pmod{p}\\\\-1\\quad&amp; otherwise\\end{aligned}\\right.\n\\end{gather*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:4.824108em;vertical-align:-2.162054em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.662054em;\"><span style=\"top:-4.6620539999999995em;\"><span class=\"pstrut\" style=\"height:4.512054em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"mspace\">&nbsp;<\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r<\/span><span class=\"mord mathnormal\">im<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\">&nbsp;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"minner\"><span class=\"mopen\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.3500199999999998em;\"><span style=\"top:-2.19999em;\"><span class=\"pstrut\" style=\"height:3.15em;\"><\/span><span class=\"delimsizinginner delim-size4\"><span>\u23a9<\/span><\/span><\/span><span style=\"top:-2.19199em;\"><span class=\"pstrut\" style=\"height:3.15em;\"><\/span><span style=\"height:0.31599999999999984em;width:0.889em;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.889em\" height=\"0.31599999999999984em\" style=\"width:0.889em\" viewBox=\"0 0 889 316\" preserveAspectRatio=\"xMinYMin\"><path d=\"M384 0 H504 V316 H384z M384 0 H504 V316 H384z\"><\/path><\/svg><\/span><\/span><span style=\"top:-3.15001em;\"><span class=\"pstrut\" style=\"height:3.15em;\"><\/span><span class=\"delimsizinginner delim-size4\"><span>\u23a8<\/span><\/span><\/span><span style=\"top:-4.292009999999999em;\"><span class=\"pstrut\" style=\"height:3.15em;\"><\/span><span style=\"height:0.31599999999999984em;width:0.889em;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.889em\" height=\"0.31599999999999984em\" style=\"width:0.889em\" viewBox=\"0 0 889 316\" preserveAspectRatio=\"xMinYMin\"><path d=\"M384 0 H504 V316 H384z M384 0 H504 V316 H384z\"><\/path><\/svg><\/span><\/span><span style=\"top:-4.600019999999999em;\"><span class=\"pstrut\" style=\"height:3.15em;\"><\/span><span class=\"delimsizinginner delim-size4\"><span>\u23a7<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.8500199999999998em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.512054em;\"><span style=\"top:-4.672054em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><\/span><\/span><span style=\"top:-3.147946em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><\/span><\/span><span style=\"top:-1.6479460000000001em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.012054em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.512054em;\"><span style=\"top:-4.672054em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">a<\/span><\/span><\/span><span style=\"top:-3.147946em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2261<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\">&nbsp;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"><\/span><span class=\"inner\"><span class=\"mord\"><span class=\"mrel\">\ue020<\/span><\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">\u2261<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mspace allowbreak\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathrm\">mod<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.3333333333333333em;\"><\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span style=\"top:-1.6479460000000001em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mord mathnormal\">o<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mord mathnormal\">h<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">er<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mord mathnormal\">se<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.012054em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.162054em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 class=\"mume-header\" id=\"underlinethm-euler\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><munder accentunder=\"true\"><mrow><mi>T<\/mi><mi>h<\/mi><mi>m<\/mi><\/mrow><mo stretchy=\"true\">\u203e<\/mo><\/munder><mtext>&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mi>u<\/mi><mi>l<\/mi><mi>e<\/mi><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\underline{Thm}\\ (Euler)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord underline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-2.84em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"underline-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">T<\/span><span class=\"mord mathnormal\">hm<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.20000000000000007em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\">&nbsp;<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">er<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/h3>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"center\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>p<\/mi><mo>\u2224<\/mo><mi>a<\/mi><mtext>\uff0c\u5219<\/mtext><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>a<\/mi><mi>p<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>a<\/mi><mfrac><mrow><mi>p<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/mfrac><\/msup><mspace><\/mspace><mspace width=\"1em\"><\/mspace><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mtext>\u2009<\/mtext><mtext>\u2009<\/mtext><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{gather*}\n    p\\nmid a\uff0c\u5219\\left(\\frac{a}{p}\\right)=(a^{\\frac{p-1}{2}}\\mod{p})\n\\end{gather*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.70003em;vertical-align:-1.1000150000000002em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.6000149999999997em;\"><span style=\"top:-3.6000150000000004em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel amsrm\">\u2224<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mord cjk_fallback\">\uff0c\u5219<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0588899999999999em;\"><span style=\"top:-3.4130000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9227em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.4623857142857144em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">p<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace allowbreak\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathrm\">mod<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1000150000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"center\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>p<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.2500em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>1<\/mn><mspace width=\"1em\"><\/mspace><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mi>p<\/mi><mtext>\u4e3a<\/mtext><mn>4<\/mn><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><mtext>\u578b<\/mtext><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo>\u2212<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mi>p<\/mi><mtext>\u4e3a<\/mtext><mn>4<\/mn><mi>k<\/mi><mo>+<\/mo><mn>3<\/mn><mtext>\u578b<\/mtext><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>2<\/mn><mi>p<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.2500em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>1<\/mn><mspace width=\"1em\"><\/mspace><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mi>p<\/mi><mtext>\u4e3a<\/mtext><mn>8<\/mn><mi>k<\/mi><mo>\u00b1<\/mo><mn>1<\/mn><mtext>\u578b<\/mtext><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo>\u2212<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mi>p<\/mi><mtext>\u4e3a<\/mtext><mn>8<\/mn><mi>k<\/mi><mo>\u00b1<\/mo><mn>3<\/mn><mtext>\u578b<\/mtext><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{gather*}\n    \\left(\\frac{-1}{p}\\right)=\\left\\{\\begin{aligned}1\\quad&amp; p\u4e3a4k+1\u578b\\\\-1\\quad&amp; p\u4e3a4k+3\u578b\\end{aligned}\\right.\\\\\n    \\left(\\frac{2}{p}\\right)=\\left\\{\\begin{aligned}1\\quad&amp; p\u4e3a8k\\pm1\u578b\\\\-1\\quad&amp; p\u4e3a8k\\pm3\u578b\\end{aligned}\\right.\n\\end{gather*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:6.600060000000001em;vertical-align:-3.0500300000000005em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.5500300000000005em;\"><span style=\"top:-5.55003em;\"><span class=\"pstrut\" style=\"height:3.75em;\"><\/span><span class=\"mord\"><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size4\">{<\/span><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.7500000000000002em;\"><span style=\"top:-3.91em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><\/span><\/span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2500000000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.7500000000000002em;\"><span style=\"top:-3.91em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mord cjk_fallback\">\u4e3a<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord cjk_fallback\">\u578b<\/span><\/span><\/span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mord cjk_fallback\">\u4e3a<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mord cjk_fallback\">\u578b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2500000000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><span style=\"top:-2.2499999999999996em;\"><span class=\"pstrut\" style=\"height:3.75em;\"><\/span><span class=\"mord\"><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size4\">{<\/span><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.7500000000000002em;\"><span style=\"top:-3.91em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><\/span><\/span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2500000000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.7500000000000002em;\"><span style=\"top:-3.91em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mord cjk_fallback\">\u4e3a<\/span><span class=\"mord\">8<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u00b1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord cjk_fallback\">\u578b<\/span><\/span><\/span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mord cjk_fallback\">\u4e3a<\/span><span class=\"mord\">8<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u00b1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mord cjk_fallback\">\u578b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2500000000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.0500300000000005em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 class=\"mume-header\" id=\"4k1%E5%9E%8B%E7%B4%A0%E6%95%B0%E6%9C%89%E6%97%A0%E7%A9%B7%E5%A4%9A%E4%B8%AA\">4k+1\u578b\u7d20\u6570\u6709\u65e0\u7a77\u591a\u4e2a<\/h3>\n\n<h3 class=\"mume-header\" id=\"%E4%BA%8C%E6%AC%A1%E4%BA%92%E5%8F%8D%E5%BE%8B\">\u4e8c\u6b21\u4e92\u53cd\u5f8b<\/h3>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"center\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>p<\/mi><mo separator=\"true\">,<\/mo><mi>q<\/mi><mtext>&nbsp;<\/mtext><mi>o<\/mi><mi>d<\/mi><mi>d<\/mi><mtext>&nbsp;<\/mtext><mi>p<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><mtext>\uff0c\u5219<\/mtext><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>p<\/mi><mi>q<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>q<\/mi><mi>p<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>p<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>4<\/mn><\/mfrac><\/msup><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{gather*}\n    p,q\\ odd\\ prime\uff0c\u5219\\left(\\frac{p}{q}\\right)\\left(\\frac{q}{p}\\right)=(-1)^\\frac{(p-1)(q-1)}{4}\n\\end{gather*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.70003em;vertical-align:-1.1000150000000002em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.6000149999999997em;\"><span style=\"top:-3.6000150000000004em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">q<\/span><span class=\"mspace\">&nbsp;<\/span><span class=\"mord mathnormal\">o<\/span><span class=\"mord mathnormal\">dd<\/span><span class=\"mspace\">&nbsp;<\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r<\/span><span class=\"mord mathnormal\">im<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord cjk_fallback\">\uff0c\u5219<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">q<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">q<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.13945em;\"><span style=\"top:-3.4130000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0377857142857143em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.5020714285714285em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">p<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">q<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1000150000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E6%8A%BD%E5%B1%89%E5%8E%9F%E7%90%86\">\u62bd\u5c49\u539f\u7406<\/h2>\n\n<h3 class=\"mume-header\" id=\"4k1%E5%9E%8B%E7%B4%A0%E6%95%B0%E4%B8%BA%E5%B9%B3%E6%96%B9%E5%92%8C\">4k+1\u578b\u7d20\u6570\u4e3a\u5e73\u65b9\u548c<\/h3>\n\n<p>\u8bc1\u660e\uff1a<br>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"center\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>p<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>\u21d2<\/mo><mi mathvariant=\"normal\">\u2203<\/mi><mi>s<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">{<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><mrow><mi>p<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mo stretchy=\"false\">}<\/mo><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><msup><mi>s<\/mi><mn>2<\/mn><\/msup><mo>\u2261<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333333333333333em\"><\/mspace><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>T<\/mi><mo>:<\/mo><mo>=<\/mo><mo stretchy=\"false\">{<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>s<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mn>0<\/mn><mo>\u2264<\/mo><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo>\u2264<\/mo><mo stretchy=\"false\">\u230a<\/mo><msqrt><mi>p<\/mi><\/msqrt><mo stretchy=\"false\">\u230b<\/mo><mo stretchy=\"false\">}<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mi>T<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mo stretchy=\"false\">\u230a<\/mo><msqrt><mi>p<\/mi><\/msqrt><mo stretchy=\"false\">\u230b<\/mo><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>&gt;<\/mo><mi>p<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi mathvariant=\"normal\">\u2203<\/mi><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">(<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2261\u0338<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2227<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><mi>s<\/mi><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>\u2261<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>+<\/mo><mi>s<\/mi><msub><mi>b<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>\u2261<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2261<\/mo><mn>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>p<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2264<\/mo><mn>2<\/mn><mo stretchy=\"false\">\u230a<\/mo><msqrt><mi>p<\/mi><\/msqrt><msup><mo stretchy=\"false\">\u230b<\/mo><mn>2<\/mn><\/msup><mo>&lt;<\/mo><mn>2<\/mn><mi>p<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo>\u21d2<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mi>p<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{gather*}\n    \\left(\\frac{-1}{p}\\right)=1\\Rightarrow\\exists s\\in\\{1,2,\\dots,{p-1}\\},\\quad s^2\\equiv -1\\pmod{p}\\\\\n    T:=\\{a+bs|0\\le a,b\\le\\lfloor\\sqrt{p}\\rfloor\\}\\\\\n    |T|=(1+\\lfloor\\sqrt{p}\\rfloor)^2&gt;p\\\\\n    \\exists a_1,b_1,a_2,b_2,\\quad (a_1,b_1)\\not \\equiv(a_2,b_2)\\wedge a_1+sb_1\\equiv a_2+sb_2\\\\\n    a_1-a_2\\equiv s(b_1-b_2)\\\\\n    (a_1-a_2)^2+(b_1-b_2)^2\\equiv 0\\\\\n    p|(a_1-a_2)^2+(b_1-b_2)^2\\le 2\\lfloor\\sqrt{p}\\rfloor^2&lt;2p\\\\\n    \\Rightarrow (a_1-a_2)^2+(b_1-b_2)^2=p\n\\end{gather*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:13.296461999999998em;vertical-align:-6.398230999999999em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:6.898230999999999em;\"><span style=\"top:-8.898231em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><span class=\"mclose\">}<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">s<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2261<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mspace allowbreak\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathrm\">mod<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.3333333333333333em;\"><\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span style=\"top:-6.808201em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">T<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">{<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mord\">\u22230<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">\u230a<\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751935em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-2.711935em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.28806499999999996em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">\u230b}<\/span><\/span><\/span><span style=\"top:-5.284093em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">T<\/span><span class=\"mord\">\u2223<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mopen\">\u230a<\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751935em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-2.711935em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.28806499999999996em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">\u230b<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-3.7840930000000004em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\">\u2203<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"><\/span><span class=\"inner\"><span class=\"mord\"><span class=\"mrel\">\ue020<\/span><\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">\u2261<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2227<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2261<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-2.2840930000000004em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2261<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><span style=\"top:-0.7599850000000008em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2261<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><span style=\"top:0.7641229999999988em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"mord\">\u2223<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mopen\">\u230a<\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751935em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:-2.711935em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.28806499999999996em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">\u230b<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><span style=\"top:2.288230999999999em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:6.398230999999999em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 class=\"mume-header\" id=\"d-%E6%9F%93%E8%89%B2%E4%BA%8C%E7%BB%B4%E6%95%B4%E7%82%B9%E4%B8%80%E5%AE%9A%E5%AD%98%E5%9C%A8%E5%9B%9B%E9%A1%B6%E7%82%B9%E5%90%8C%E8%89%B2%E7%9F%A9%E5%BD%A2\">d-\u67d3\u8272\u4e8c\u7ef4\u6574\u70b9\u4e00\u5b9a\u5b58\u5728\u56db\u9876\u70b9\u540c\u8272\u77e9\u5f62<\/h3>\n\n<h2 class=\"mume-header\" id=\"ramsey-theory\">Ramsey Theory<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>n<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u22ef<\/mo><mtext>\u2009<\/mtext><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mi>r<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>&lt;<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R(n_1,n_2,\\cdots,n_r)&lt;\\infty<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord\">\u221e<\/span><\/span><\/span><\/span><\/span><br>\n\u8bc1\u660e\uff1a<br>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>n<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mi>r<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2264<\/mo><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>n<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mi>r<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>+<\/mo><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>n<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mi>r<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>+<\/mo><mo>\u22ef<\/mo><mo>+<\/mo><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>n<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>n<\/mi><mi>r<\/mi><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{align*}\n    R(n_1,n_2,\\dots,n_{r})-1\\le&amp; R(n_1,n_2,\\dots,n_{r})-1\n    \\\\&amp;+R(n_1,n_2-1,\\dots,n_{r})-1\\\\\n    &amp;+\\cdots+R(n_1,n_2,\\dots,n_{r}-1)-1\\\\\n    &amp;+1\n\\end{align*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:6em;vertical-align:-2.7500000000000004em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.25em;\"><span style=\"top:-5.41em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><\/span><\/span><span style=\"top:-3.91em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><\/span><\/span><span style=\"top:-2.4099999999999993em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><\/span><\/span><span style=\"top:-0.9099999999999997em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.7500000000000004em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.25em;\"><span style=\"top:-5.41em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-3.91em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-2.4099999999999993em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-0.9099999999999997em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.7500000000000004em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"center\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>6<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>9<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{gather*}\n    R(3,3)=6\\\\\n    R(3,4)=9\n\\end{gather*}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:3.0000000000000004em;vertical-align:-1.2500000000000002em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.7500000000000002em;\"><span style=\"top:-3.91em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">6<\/span><\/span><\/span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">4<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2500000000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"schur-theorem\">Schur Theorem<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2203<\/mi><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mtext>\u5bf9<\/mtext><mo stretchy=\"false\">{<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mo stretchy=\"false\">}<\/mo><mtext>&nbsp;<\/mtext><mi>d<\/mi><mo>\u2212<\/mo><mtext>\u67d3\u8272<\/mtext><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi mathvariant=\"normal\">\u2203<\/mi><mtext>\u540c\u8272<\/mtext><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo separator=\"true\">,<\/mo><mi>z<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mi>z<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\exists N(d),\\quad \u5bf9\\{1,2,\\dots,{N(d)}\\}\\ d-\u67d3\u8272,\\quad\\exists \u540c\u8272x,y,z,\\quad x+y=z<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord cjk_fallback\">\u5bf9<\/span><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mclose\">)<\/span><\/span><span class=\"mclose\">}<\/span><span class=\"mspace\">&nbsp;<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"><\/span><span class=\"mord cjk_fallback\">\u67d3\u8272<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">\u2203<\/span><span class=\"mord cjk_fallback\">\u540c\u8272<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.04398em;\">z<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.04398em;\">z<\/span><\/span><\/span><\/span><\/span><br>\n\u8bc1\u660e\u601d\u8def\uff1a<br>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>\u6309<\/mtext><mi mathvariant=\"normal\">\u2223<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>j<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mtext>\u7684\u67d3\u8272\u4e3a<\/mtext><msub><mi>v<\/mi><mi>i<\/mi><\/msub><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mtext>\u8fb9\u67d3\u8272\uff0c\u5229\u7528\u62c9\u59c6\u585e\u7406\u8bba\u5f97\u5230\u540c\u8272\u4e09\u89d2\u5f62<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\u6309|i-j|\u7684\u67d3\u8272\u4e3av_iv_j\u8fb9\u67d3\u8272\uff0c\u5229\u7528\u62c9\u59c6\u585e\u7406\u8bba\u5f97\u5230\u540c\u8272\u4e09\u89d2\u5f62<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord cjk_fallback\">\u6309<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.036108em;vertical-align:-0.286108em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.05724em;\">j<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord cjk_fallback\">\u7684\u67d3\u8272\u4e3a<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.311664em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord cjk_fallback\">\u8fb9\u67d3\u8272\uff0c\u5229\u7528\u62c9\u59c6\u585e\u7406\u8bba\u5f97\u5230\u540c\u8272\u4e09\u89d2\u5f62<\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"van-der-waerden-theorem\">Van der Waerden Theorem<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2203<\/mi><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>d<\/mi><mo separator=\"true\">,<\/mo><mi>l<\/mi><mo stretchy=\"false\">)<\/mo><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mtext>\u5bf9<\/mtext><mo stretchy=\"false\">{<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>d<\/mi><mo separator=\"true\">,<\/mo><mi>l<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mo stretchy=\"false\">}<\/mo><mtext>&nbsp;<\/mtext><mi>d<\/mi><mo>\u2212<\/mo><mtext>\u67d3\u8272<\/mtext><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi mathvariant=\"normal\">\u2203<\/mi><mtext>\u540c\u8272<\/mtext><mi>l<\/mi><mo>\u2212<\/mo><mi>A<\/mi><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>l<\/mi><mo>\u2212<\/mo><mtext>\u957f\u7b49\u5dee\u6570\u5217<\/mtext><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\exists W(d,l),\\quad \u5bf9\\{1,2,\\dots,{W(d,l)}\\}\\ d-\u67d3\u8272,\\quad\\exists \u540c\u8272l-AP(l-\u957f\u7b49\u5dee\u6570\u5217)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord cjk_fallback\">\u5bf9<\/span><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l<\/span><span class=\"mclose\">)<\/span><\/span><span class=\"mclose\">}<\/span><span class=\"mspace\">&nbsp;<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"><\/span><span class=\"mord cjk_fallback\">\u67d3\u8272<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">\u2203<\/span><span class=\"mord cjk_fallback\">\u540c\u8272<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord cjk_fallback\">\u957f\u7b49\u5dee\u6570\u5217<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><br>\n\u8bc1\u660e\u5173\u952e\uff1a<br>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math 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id=\"%E5%B9%B3%E9%9D%A2%E6%95%B4%E7%82%B92-%E6%9F%93%E8%89%B2%E4%B8%80%E5%AE%9A%E5%AD%98%E5%9C%A8%E9%A1%B6%E7%82%B9%E5%90%8C%E8%89%B2%E7%AD%89%E8%85%B0%E7%9B%B4%E8%A7%92%E4%B8%89%E8%A7%92%E5%BD%A2\">\u5e73\u9762\u6574\u70b92-\u67d3\u8272\u4e00\u5b9a\u5b58\u5728\u9876\u70b9\u540c\u8272\u7b49\u8170\u76f4\u89d2\u4e09\u89d2\u5f62<\/h2>\n\n<h2 class=\"mume-header\" id=\"%E5%B9%B3%E9%9D%A2%E7%82%B92-%E6%9F%93%E8%89%B2%E4%B8%80%E5%AE%9A%E5%AD%98%E5%9C%A8%E4%B8%8E%E7%BB%99%E5%AE%9A30%E5%BA%A6%E8%A7%92%E7%9B%B4%E8%A7%92%E4%B8%89%E8%A7%92%E5%BD%A2%E5%85%A8%E7%AD%89%E7%9A%84%E9%A1%B6%E7%82%B9%E5%90%8C%E8%89%B2%E4%B8%89%E8%A7%92%E5%BD%A2\">\u5e73\u9762\u70b92-\u67d3\u8272\u4e00\u5b9a\u5b58\u5728\u4e0e\u7ed9\u5b9a30\u5ea6\u89d2\u76f4\u89d2\u4e09\u89d2\u5f62\u5168\u7b49\u7684\u9876\u70b9\u540c\u8272\u4e09\u89d2\u5f62<\/h2>\n\n<h2 class=\"mume-header\" 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