{"id":565,"date":"2021-04-07T12:00:00","date_gmt":"2021-04-07T04:00:00","guid":{"rendered":"https:\/\/blog.cruciferslab.net\/?p=565"},"modified":"2021-04-06T05:05:48","modified_gmt":"2021-04-05T21:05:48","slug":"project-euler-%e8%a7%a3%e9%a1%8c%e7%ad%86%e8%a8%98-2","status":"publish","type":"post","link":"https:\/\/blog.cruciferslab.net\/?p=565","title":{"rendered":"Project Euler \u89e3\u984c\u7b46\u8a18 (2)"},"content":{"rendered":"\n

#152 Writing 1\/2 as a sum of inverse squares<\/a> (\u96e3\u5ea6 65%)<\/h2>\n\n\n\n

\u9019\u984c\u8eba\u5728\u6211\u7684\u672a\u89e3\u984c\u88e1\u8eba\u5f88\u4e45\u4e86\uff1a\u6709\u4e00\u9663\u5b50\u6211\u5728\u5617\u8a66\u5f9e\u982d\u958b\u59cb\u7b54\u984c\uff0c\u5beb\u4e00\u5beb\u5c31\u649e\u4e0a\u9019\u984c\u9019\u500b\u8edf\u7246\uff0c\u6240\u4ee5\u5c31\u958b\u59cb\u5f80\u5f8c\u4e82\u8df3\u984c\u76ee XD \u4e5f\u5c31\u662f\u8aaa\uff0c\u9664\u4e86\u5728\u7d04\u4e09\u56db\u767e\u984c\u90a3\u9644\u8fd1\u6642\u6709\u6bd4\u8f03\u96c6\u4e2d\u5728\u4e00\u51fa\u984c\u5c31\u53bb\u89e3\u4e4b\u5916\uff0c\u5176\u4ed6\u7684\u89e3\u984c\u9032\u5ea6\u90fd\u662f\u9019\u6a23\u6563\u6563\u7684\uff0c\u770b\u5230\u6709\u9ede\u96e3\u7684\u984c\u76ee\u5c31\u7d66\u5b83\u62fc\u547d\u8df3\uff0c\u6240\u4ee5\u9019\u984c\u624d\u7559\u8457\u9019\u9ebc\u4e45\u3002 <\/p>\n\n\n\n

\u984c\u76ee\u5f88\u7c21\u55ae\uff1a\u5c31\u5982\u540c\u6a19\u984c\u8aaa\u7684\uff0c\u628a 1\/2 \u5beb\u6210\u4e0d\u540c\u6574\u6578\u5e73\u65b9\u7684\u5012\u6578\u548c\u3002\u984c\u76ee\u6558\u8ff0\u7d66\u4e86\u4e00\u500b\u7bc4\u4f8b\uff1a<\/p>\n\n\n\n

$$\\begin{align}\\dfrac{1}{2} &= \\dfrac{1}{2^2} + \\dfrac{1}{3^2} + \\dfrac{1}{4^2} + \\dfrac{1}{5^2} +\\\\
&\\quad \\dfrac{1}{7^2} + \\dfrac{1}{12^2} + \\dfrac{1}{15^2} + \\dfrac{1}{20^2} +\\\\
&\\quad \\dfrac{1}{28^2} + \\dfrac{1}{35^2}\\end{align}$$<\/p>\n\n\n\n

\u4e26\u4e14\u7d66\u51fa\u5982\u679c\u53ea\u8003\u616e 2 \u5230 45 \u7684\u5e73\u65b9\u5012\u6578\u7684\u8a71\uff0c\u9023\u540c\u9019\u500b\u7bc4\u4f8b\u4e00\u5171\u53ea\u6709\u4e09\u7d44\u89e3\u3002\u984c\u76ee\u8981\u554f\u7684\u662f\uff1a\u5982\u679c\u653e\u5927\u7bc4\u570d\u5230 2 \u5230 80 \u7684\u8a71\u6709\u5e7e\u7d44\u89e3\uff1f<\/p>\n\n\n\n

\u4e4b\u6240\u4ee5\u662f\u8edf\u7246\u662f\u56e0\u70ba\uff0c2 \u5230 80 \u9019\u500b\u7bc4\u570d\u5f88\u5fae\u5999\u7684\u7a0d\u5fae\u5927\u4e86\u4e00\u9ede\uff1b\u9019\u500b\u984c\u76ee\u672c\u8cea\u4e0a\u662f\u4e00\u500b\u5b50\u96c6\u548c\u554f\u984c<\/a>\uff0c\u8981\u6211\u5011\u5728\u7d66\u5b9a\u7684 79 \u500b\u5143\u7d20\u4e2d\u9078\u51fa\u6578\u500b\u4f86\u52a0\u5230\u76ee\u6a19 1\/2\uff1b\u7531\u65bc\u5b50\u96c6\u548c\u554f\u984c\u662f\u8457\u540d\u7684 NP-Complete \u554f\u984c\uff0c\u6240\u9700\u8981\u7684\u5de5\u4f5c\u91cf\u4e0d\u6703\u6bd4\u628a 79 \u500b\u6578\u500b\u5225\u9078\u4e0d\u9078\u6240\u5f97\u5230\u7684 \\(2^{79}\\) \u7a2e\u9078\u9805\u9010\u4e00\u5617\u8a66\u4f86\u5f97\u597d\u591a\u5c11\u3002<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

<\/p>\n\n\n\n\n\n\n\n

\u5982\u679c\u4f7f\u7528\u9019\u4e9b\u6578\u5b57\u7684\u6578\u5b78\u6027\u8cea\u5462\uff1f\u81ea\u7136\u6578\u5e73\u65b9\u7684\u5012\u6578\u548c\u4e5f\u662f\u500b\u5f88\u6709\u540d\u7684\u554f\u984c\uff0c\u53eb\u505a\u5df4\u585e\u723e\u554f\u984c<\/a>\uff0c\u5176\u7e3d\u548c\u7d04\u662f 1.644934\uff0c\u4e5f\u5c31\u662f\u7531 2 \u5230\u7121\u7aae\u7684\u5e73\u65b9\u5012\u6578\u548c\u662f\u9019\u548c\u6e1b\u53bb 1 \u7d04\u662f 0.644934\uff0c\u548c\u6211\u5011\u7684\u76ee\u6a19 0.5 \u7684\u5dee\u8ddd\u53c8\u96e2\u7684\u6709\u4e00\u9ede\u9060\uff0c\u4e0d\u592a\u80fd\u5920\u63d0\u524d\u904e\u6ffe\u9078\u9805\uff1b\u80fd\u7c21\u55ae\u63a8\u5f97\u7684\u53ea\u6709 \\(\\frac1{2^2}\\) \u4e0d\u80fd\u4e0d\u9078\uff0c\u4ee5\u53ca \\(\\frac1{3^2}\\) \u8ddf \\(\\frac1{4^2}\\) \u4e0d\u80fd\u4e00\u8d77\u4e0d\u9078\uff1b\u8981\u518d\u4e0b\u53bb\u5c31\u4e0d\u5bb9\u6613\u63a8\u689d\u4ef6\u4e86\u3002\u4f3c\u4e4e\u9084\u662f\u9700\u8981\u6df1\u6316\u4e00\u9ede\u5176\u4ed6\u7684\u6578\u5b78\u689d\u4ef6\u4f86\u4f7f\u7528\u3002<\/p>\n\n\n\n

\u7a81\u7834\u53e3\uff1a\u52a0\u8d77\u4f86\u5427\uff01<\/h2>\n\n\n\n

\u8aaa\u8d77\u4f86\uff0c\u984c\u76ee\u7d66\u7684\u9019\u7d44\u7b54\u6848\u5f88\u6709\u8da3\uff1a\u4e2d\u9593\u751a\u81f3\u51fa\u73fe\u4e86 28 \u548c 35 \u9019\u7a2e 7 \u7684\u500d\u6578\u5206\u6bcd\uff0c\u4f46\u5168\u90e8\u52a0\u8d77\u4f86\u4e4b\u5f8c\u537b\u80fd\u901a\u901a\u6d88\u5149\u5149\uff0c\u6700\u5f8c\u5206\u6bcd\u53ea\u5269\u4e0b\u4e00\u500b 2\u3002\u5230\u5e95\u9019\u662f\u600e\u9ebc\u6d88\u7684\uff1f<\/p>\n\n\n\n

\u5bb9\u6613\u770b\u5230\u9019\u7d44\u89e3\u88e1\uff0c\u5206\u6bcd\u662f 7 \u7684\u500d\u6578\u7684\u6709\u4e09\u500b\uff1a\\(\\frac1{7^2},\\ \\frac1{28^2},\\ \\frac1{35^2}\\)\uff0c\u800c\u5b83\u5011\u52a0\u8d77\u4f86\u662f\uff1a<\/p>\n\n\n\n

$$\\frac1{7^2}+\\frac1{28^2}+\\frac1{35^2}=\\frac{20^2+5^2+4^2}{140^2}=\\frac{441}{140^2}=\\frac{9\\times7^2}{140^2}=\\frac9{20^2}$$<\/p>\n\n\n\n

\u65bc\u662f\u6211\u5011\u767c\u73fe\u4e86\uff1a\u539f\u4f86\u662f\u56e0\u70ba\u901a\u5206\u76f8\u52a0\u4e4b\u5f8c\uff0c\u5206\u5b50\u51fa\u73fe\u4e86 \\(7^2\\) \u7684\u500d\u6578\uff0c\u56e0\u6b64\u53ef\u4ee5\u628a\u901a\u5206\u5f8c\u5206\u6bcd\u7684 \\(7^2\\) \u7d66\u6d88\u6389\u3002\u9019\u500b\u689d\u4ef6\u5176\u5be6\u5f88\u5927\u5730\u9650\u5236\u4e86\u4e00\u4e9b\u5927\u56e0\u6578\u7684\u51fa\u5834\u6a5f\u6703\uff1a\u4f8b\u5982 31\uff0c\u5728 2 \u5230 80 \u7684\u7bc4\u570d\u5167\u7684 31 \u7684\u500d\u6578\u53ea\u6709 31 \u548c 62\uff0c\u5e73\u65b9\u5012\u6578\u548c\u4e0d\u6703\u52a0\u51fa \\(31^2\\) \u7684\u500d\u6578\u51fa\u4f86\uff0c\u56e0\u6b64\u5982\u679c\u5b83\u5011\u4e00\u51fa\u5834\uff0c\u5206\u6bcd\u7684 \\(31^2\\) \u6c92\u6709\u6a5f\u6703\u6d88\u6389\uff0c\u7d50\u679c\u4e5f\u5c31\u4e0d\u53ef\u80fd\u662f 1\/2 \u4e86\u3002<\/p>\n\n\n\n

\u7167\u7406\u4f86\u8aaa\uff0c\u5c0d\u5947\u8cea\u6578\u4f86\u8aaa\u6240\u9078\u7684\u8a72\u5947\u8cea\u6578\u7684\u500d\u6578\u4e4b\u5e73\u65b9\u5012\u6578\u548c\u90fd\u8981\u6709\u9019\u500b\u6027\u8cea\uff0c\u4f46\u9019\u6a23\u4e00\u4f86\u5408\u6578\u5230\u5e95\u8981\u7d66\u8ab0\u7b97\uff1f\u5982\u679c\u8981\u6b78\u7d66\u6700\u5927\u8cea\u56e0\u6578\u7684\u8a71\uff0c\u90a3\u5c0f\u4e00\u9ede\u7684\u8cea\u56e0\u6578\u5c31\u6709\u53ef\u80fd\u6703\u52a0\u5165\u4e00\u4e9b\u548c\u5176\u4ed6\u4eba\u9577\u5f97\u4e0d\u592a\u4e00\u6a23\u7684\u503c\uff0c\u9019\u6a23\u641c\u5c0b\u4e5f\u5f88\u9ebb\u7169\u3002\u4ee5\u4e0a\u9762\u70ba\u4f8b\uff0c\u5982\u679c\u63a5\u4e0b\u4f86\u8003\u616e 5 \u7684\u500d\u6578\u7684\u8a71\uff0c\u9019\u88e1\u5c07\u6703\u662f\uff1a<\/p>\n\n\n\n

$$\\frac1{5^2}+\\frac1{15^2}+\\frac1{20^2}+\\frac9{20^2}=\\frac{12^2+4^2+3^2+9\\times3^2}{60^2}=\\frac{250}{60^2}=\\frac{10}{12^2}=\\frac5{72}$$<\/p>\n\n\n\n

\u9019\u500b \\(\\frac5{72}\\) \u5c31\u4e0d\u50cf\u4e0a\u9762\u7684\u7d50\u679c\u4e00\u6a23\u662f\u500b\u6709\u7406\u6578\u5b8c\u5168\u5e73\u65b9\u3002\u96d6\u7136 5 \u7684\u56e0\u6578\u6d88\u6389\u4e86\uff0c\u4f46\u56e0\u70ba\u5f62\u5f0f\u8b8a\u5316\u592a\u5927\uff0c\u9019\u4e26\u4e0d\u5bb9\u6613\u6df7\u9032 3 \u7684\u4e00\u822c\u8a0e\u8ad6\u88e1\u6c42\u7b97\u3002<\/p>\n\n\n\n

\u5927\u8cea\u56e0\u6578\u641c\u5c0b<\/h2>\n\n\n\n

\u4e0d\u904e\uff0c\u81f3\u5c11\u5c0d\u50cf\u4e0a\u9762\u8209\u7684 31 \u9019\u7a2e\u5927\u4e00\u9ede\u7684\u8cea\u56e0\u6578\u6211\u5011\u53ef\u4ee5\u5229\u7528\u9019\u500b\u6027\u8cea\u52a0\u4ee5\u6392\u9664\u3002\u5f88\u81ea\u7136\u7684\u554f\u984c\u5c31\u662f\u9084\u6709\u6c92\u6709\u4e00\u4e9b\u50cf\u9019\u6a23\u7684\u7d44\u5408\uff1f\u9019\u6a23\u4e00\u4f86\u6211\u5011\u53ea\u9700\u8981\u5c0d\u53ea\u6709\u5c0f\u8cea\u56e0\u6578\u7684\u6578 (\u9019\u7a2e\u6578\u88ab\u7a31\u70ba\u5149\u6ed1\u6578<\/a>) \u53bb\u641c\u5c0b\u5373\u53ef\u3002\u7d50\u679c\u5f88\u6709\u8da3\uff1a\u5982\u679c\u8003\u616e\u6700\u5927\u8cea\u56e0\u6578\u5927\u65bc\u7b49\u65bc 11 \u7684\u6578\u4e4b\u7d44\u5408\u7684\u8a71\uff0c\u5728\u984c\u76ee\u554f\u7684 2 \u5230 80 \u7684\u7bc4\u570d\u4e2d\u53ea\u6709\u4e00\u500b\u7d44\u5408\u6709\u6211\u5011\u8981\u7684\u6027\u8cea\uff1a<\/p>\n\n\n\n

$$\\frac1{13^2}+\\frac1{39^2}+\\frac1{52^2}=\\frac{12^2+4^2+3^2}{156^2}=\\frac{169}{156^2}=\\frac1{12^2}$$<\/p>\n\n\n\n

\u9019\u500b\u5206\u6bcd\u751a\u81f3\u9023 5 \u548c 7 \u90fd\u6c92\u6709\uff0c\u6240\u4ee5\u6211\u5011\u53ef\u4ee5\u9032\u4e00\u6b65\u641c\u5c0b\u6700\u5927\u56e0\u6578\u662f 7 \u7684\u7d44\u5408\uff0c\u4e0d\u6703\u53d7\u9019\u4e00\u7d44\u5230\u5e95\u9078\u4e0d\u9078\u5c0d\u641c\u5c0b\u7684\u5f71\u97ff\u3002\u4ee5\u4e0b\u5c31\u662f\u7bc4\u570d\u5167\u80fd\u5920\u4f7f\u7528\u7684 7 \u7684\u7d44\u5408\uff0c\u4e00\u5171 19 \u7d44\uff1a<\/p>\n\n\n\n

\n
\n
  • {28, 35, 42, 56, 63, 70} = 89\/25920<\/li>
  • {21, 42, 63} = 1\/324<\/li>
  • {21, 35, 56} = 49\/14400<\/li>
  • {14, 56, 70} = 9\/1600<\/li>
  • {14, 35, 56, 63} = 841\/129600<\/li>
  • {14, 28, 42} = 1\/144<\/li>
  • {14, 21, 42, 56, 63, 70} = 1129\/129600<\/li>
  • {14, 21, 35, 63, 70} = 7\/810<\/li>
  • {14, 21, 28, 35, 42, 56} = 149\/14400<\/li><\/ul>\n<\/div>\n\n\n\n
    \n
    • {7, 63, 70} = 169\/8100<\/li>
    • {7, 28, 42, 56} = 13\/576<\/li>
    • {7, 28, 35} = 9\/400<\/li>
    • {7, 21, 35, 56, 63, 70} = 629\/25920<\/li>
    • {7, 21, 28, 42, 70} = 89\/3600<\/li>
    • {7, 21, 28, 35, 42, 63} = 829\/32400<\/li>
    • {7, 14, 28, 42, 63, 70} = 901\/32400<\/li>
    • {7, 14, 28, 35, 56, 70} = 9\/320<\/li>
    • {7, 14, 21} = 1\/36<\/li>
    • {7, 14, 21, 28, 35, 42, 56, 63, 70} = 809\/25920<\/li><\/ul>\n<\/div>\n<\/div>\n\n\n\n

      \u9019\u88e1\u7684 19 \u7d44\u7684\u5206\u6bcd\u5f88\u591a\u90fd\u6709\u5e36\u6578\u500b 5\uff0c\u6240\u4ee5\u4e0d\u5bb9\u6613\u4ee3\u5165 5 \u7684\u500d\u6578\u8a0e\u8ad6\u3002\u4e0d\u904e\u5230\u9019\u88e1\u81f3\u5c11\u6211\u5011\u7be9\u5230\u5269\u4e0b\u6240\u6709\u7684 5-\u5149\u6ed1\u6578\u4e86\uff1b\u5c0f\u65bc\u7b49\u65bc 80 \u7684 5-\u5149\u6ed1\u6578\u53ea\u6709 29 \u500b\uff0c\u6bd4\u8d77\u539f\u4f86\u7684 79 \u500b\u5c11\u592a\u591a\u4e86\u3002<\/p>\n\n\n\n

      \u518d\u6df1\u5165\u601d\u8003\u5c31\u6703\u767c\u73fe\uff0c\u6211\u5011\u9019\u6a23\u505a\u5176\u5be6\u5df2\u7d93\u628a\u4e00\u500b 79 \u5143\u7d20\u7684\u5b50\u96c6\u548c\u554f\u984c\uff0c\u8f49\u8b8a\u6210\u4e86 40 \u500b 29 \u5143\u7d20\u7684\u5b50\u96c6\u548c\u554f\u984c\u300240 \u500b\u662f\u56e0\u70ba\uff0c7 \u7684\u9078\u9805\u53ef\u4ee5\u9078 19 \u7a2e\u7d44\u5408\u6216\u4e0d\u9078\uff0c\u7136\u5f8c\u4e0a\u9762 13 \u7684\u90a3\u7d44\u4e5f\u80fd\u9078\u548c\u4e0d\u9078\uff0c\u7e3d\u8a08\u5c31\u6709 40 \u500b\u76ee\u6a19\uff1b\u7531\u65bc\u9019 40 \u500b\u76ee\u6a19\u5728\u554f\u7684\u6bcd\u96c6\u5408\u662f\u540c\u4e00\u500b 29 \u5143\u7d20\u96c6\u5408\uff0c\u6211\u5011\u53ef\u4ee5\u7528\u540c\u6a23\u7684\u52d5\u614b\u898f\u5283\u89e3\u6cd5\u5168\u90e8\u8655\u7406\u4e00\u904d\u4e4b\u5f8c\uff0c\u8a62\u554f\u9019 40 \u500b\u76ee\u6a19\u6709\u6c92\u6709\u4eba\u6709\u8e29\u4e2d\u5c31\u884c\u4e86\u3002\u5206\u6578\u7684\u8655\u7406\u5247\u53ef\u4ee5\u7c21\u55ae\u5730\u628a\u6240\u6709\u8981\u8655\u7406\u7684\u5206\u6578\u5168\u90e8\u901a\u5206\u6210\u9019 29 \u500b\u6578\u7684\u6700\u5c0f\u516c\u500d\u6578 \\((2^6\\cdot3^3\\cdot5^2=64\\cdot27\\cdot25=43200)\\) \u7684\u5e73\u65b9\u5373\u53ef\uff1a<\/p>\n\n\n\n

      \/* Final target:\n\n1\/2 = 933120000\/43200^2\n\n*\/\n\nconst int64_t half = 933120000;\n\n\/* 7 and 13 terms:\n\n{21, 42, 63}\t\t\t\t\t\t 5760000\/43200^2\n{21, 35, 56}\t\t\t\t\t\t 6350400\/43200^2\n{28, 35, 42, 56, 63, 70}\t\t\t 6408000\/43200^2\n{14, 56, 70}\t\t\t\t\t\t10497600\/43200^2\n{14, 35, 56, 63}\t\t\t\t\t12110400\/43200^2\n{14, 28, 42}\t\t\t\t\t\t12960000\/43200^2\n{14, 21, 35, 63, 70}\t\t\t\t16128000\/43200^2\n{14, 21, 42, 56, 63, 70}\t\t\t16257600\/43200^2\n{14, 21, 28, 35, 42, 56}\t\t\t19310400\/43200^2\n{7, 63, 70}\t\t\t\t\t\t\t38937600\/43200^2\n{7, 28, 35}\t\t\t\t\t\t\t41990400\/43200^2\n{7, 28, 42, 56}\t\t\t\t\t\t42120000\/43200^2\n{7, 21, 35, 56, 63, 70}\t\t\t\t45288000\/43200^2\n{7, 21, 28, 42, 70}\t\t\t\t\t46137600\/43200^2\n{7, 21, 28, 35, 42, 63}\t\t\t\t47750400\/43200^2\n{7, 14, 21}\t\t\t\t\t\t\t51840000\/43200^2\n{7, 14, 28, 42, 63, 70}\t\t\t\t51897600\/43200^2\n{7, 14, 28, 35, 56, 70}\t\t\t\t52488000\/43200^2\n{7, 14, 21, 28, 35, 42, 56, 63, 70}\t58248000\/43200^2\n{13, 39, 52}\t\t\t\t\t\t12960000\/43200^2\n\n*\/\n\nconst int64_t sevens[] = {\n       0,\n 5760000,  6350400,  6408000, 10497600, 12110400, 12960000, \n16128000, 16257600, 19310400, 38937600, 41990400, 42120000, \n45288000, 46137600, 47750400, 51840000, 51897600, 52488000, \n58248000\n};\n\nconstexpr int SEVENS_COUNT = _countof(sevens);\n\nconst int64_t thirteens[] = {0, 12960000};\n\nconstexpr int THIRTEENS_COUNT = _countof(thirteens);\n\n\/* 5-smooth numbers:\n\n1\/80^2 =   291600\/43200^2\n1\/75^2 =   331776\/43200^2\n1\/72^2 =   360000\/43200^2\n1\/64^2 =   455625\/43200^2\n1\/60^2 =   518400\/43200^2\n1\/54^2 =   640000\/43200^2\n1\/50^2 =   746496\/43200^2\n1\/48^2 =   810000\/43200^2\n1\/45^2 =   921600\/43200^2\n1\/40^2 =  1166400\/43200^2\n1\/36^2 =  1440000\/43200^2\n1\/32^2 =  1822500\/43200^2\n1\/30^2 =  2073600\/43200^2\n1\/27^2 =  2560000\/43200^2\n1\/25^2 =  2985984\/43200^2\n1\/24^2 =  3240000\/43200^2\n1\/20^2 =  4665600\/43200^2\n1\/18^2 =  5760000\/43200^2\n1\/16^2 =  7290000\/43200^2\n1\/15^2 =  8294400\/43200^2\n1\/12^2 = 12960000\/43200^2\n1\/10^2 = 18662400\/43200^2\n1\/9^2 =  23040000\/43200^2\n1\/8^2 =  29160000\/43200^2\n1\/6^2 =  51840000\/43200^2\n1\/5^2 =  74649600\/43200^2\n1\/4^2 = 116640000\/43200^2\n1\/3^2 = 207360000\/43200^2\n1\/2^2 = 466560000\/43200^2\n\n*\/\n\nconst int64_t fivesmooth[] = {\n   291600,    331776,    360000,    455625,    518400,    640000, \n   746496,    810000,    921600,   1166400,   1440000,   1822500, \n  2073600,   2560000,   2985984,   3240000,   4665600,   5760000, \n  7290000,   8294400,  12960000,  18662400,  23040000,  29160000, \n 51840000,  74649600, 116640000, 207360000, 466560000\n};<\/pre>\n\n\n\n
      \u7576\u7136\u5be6\u969b\u8655\u7406\u4e26\u6c92\u6709\u771f\u7684\u958b\u4e00\u500b\u9577\u5ea6\u70ba \\(43200^2\/2\\) \u7684\u9663\u5217\u5728\u90a3\u88e1\u4e00\u76f4\u7b97 0\uff0c\u800c\u662f\u5c31\u53ea\u8a18 DP \u9663\u5217\u88e1\u54ea\u500b\u4f4d\u7f6e\u6709\u503c\u518d\u53bb\u63a8\u9032\uff0c\u4f46\u6982\u5ff5\u662f\u4e00\u6a23\u7684\u3002\u9019\u88e1\u9084\u80fd\u56e0\u70ba\u6211\u5011\u8981\u554f\u7684\u76ee\u6a19\u6700\u5927\u5c31\u662f 1\/2\uff0c\u6240\u4ee5\u6240\u6709\u8d85\u904e\u5b83\u7684\u7e3d\u548c\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002<\/p>\n\n\n\n
      \u9019\u6a23\u5beb\u51fa\u4f86\u7684\u7a0b\u5f0f\u5728\u6211\u7684\u96fb\u8166\u4e0a\u5dee\u4e0d\u591a\u8e29\u5728 Project Euler \u7684\u300c\u4e00\u5206\u9418\u898f\u5247\u300d\u7dda\u4e0a\uff0c\u6240\u4ee5\u61c9\u8a72\u7b97\u662f\u52c9\u5f37\u904e\u95dc\u7684\u7a0b\u5f0f\u78bc\u4e86\u3002\u7a0d\u52a0\u6539\u5beb\uff0c\u82b1\u4e45\u4e00\u9ede\u7684\u6642\u9593\u662f\u53ef\u4ee5\u8dd1\u51fa\u6240\u6709\u7684\u89e3\u500b\u5225\u662f\u4ec0\u9ebc\uff0c\u9019\u88e1\u5c31\u5217\u51fa\u7bc4\u570d\u5167\u5169\u7d44\u9078\u64c7\u6578\u76ee\u6700\u591a\u7684\u89e3 (\u9078\u4e86 24 \u9805) \u505a\u70ba\u7d50\u675f\u5427\uff1a<\/p>\n\n\n\n
      $$\\begin{align}\\dfrac{1}{2} &= \\dfrac{1}{2^2} + \\dfrac{1}{3^2} + \\dfrac{1}{5^2} + \\dfrac{1}{6^2} + \\dfrac{1}{7^2} + \\dfrac{1}{8^2} + \\dfrac{1}{10^2} + \\dfrac{1}{13^2} + \\dfrac{1}{14^2} + \\dfrac{1}{18^2} + \\dfrac{1}{21^2} + \\dfrac{1}{24^2} + \\\\
      &\\quad \\dfrac{1}{28^2} + \\dfrac{1}{30^2} + \\dfrac{1}{35^2} + \\dfrac{1}{39^2} + \\dfrac{1}{40^2} + \\dfrac{1}{42^2} +  \\dfrac{1}{45^2} + \\dfrac{1}{52^2} + \\dfrac{1}{56^2} + \\dfrac{1}{60^2} + \\dfrac{1}{63^2} + \\dfrac{1}{70^2}\\\\
      &= \\dfrac{1}{2^2} + \\dfrac{1}{3^2} + \\dfrac{1}{5^2} + \\dfrac{1}{6^2} + \\dfrac{1}{7^2} + \\dfrac{1}{9^2} + \\dfrac{1}{10^2} + \\dfrac{1}{13^2} + \\dfrac{1}{14^2} + \\dfrac{1}{15^2} + \\dfrac{1}{20^2} + \\dfrac{1}{21^2} + \\\\
      &\\quad \\dfrac{1}{24^2} + \\dfrac{1}{28^2} + \\dfrac{1}{30^2} + \\dfrac{1}{35^2} + \\dfrac{1}{39^2} + \\dfrac{1}{40^2} + \\dfrac{1}{42^2} + \\dfrac{1}{52^2} + \\dfrac{1}{56^2} + \\dfrac{1}{63^2} + \\dfrac{1}{70^2} + \\dfrac{1}{72^2}
      \\end{align}$$<\/p>\n","protected":false},"excerpt":{"rendered":"
      #152 Writing 1\/2 as a sum of inverse squares (\u96e3\u5ea6 65%) \u9019 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8,3,4],"tags":[],"class_list":["post-565","post","type-post","status-publish","format-standard","hentry","category-projecteuler","category-math","category-programming"],"_links":{"self":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/565","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=565"}],"version-history":[{"count":19,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/565\/revisions"}],"predecessor-version":[{"id":588,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/565\/revisions\/588"}],"wp:attachment":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=565"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=565"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=565"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}