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[数论] 正整数解

求不大于2019的n的个数,使得方程$x^3+y^3=z^n$有正整数解$(x,y,z)$
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本帖最后由 青青子衿 于 2019-5-22 20:32 编辑

回复 1# Shiki
  1. text[x_] := GCD @@ FactorInteger[x][[All, 2]] >= 2;
  2. list = SortBy[Table[{i, j, i^3 + j^3}, {i, 1, 1000}, {j, 1, 1000}]~Flatten~1, Last];
  3. Select[list, text[#[[3]]] &] // AbsoluteTiming
复制代码
...
{"1^3 + 2^3 = 3^2"},
{"2^3 + 2^3 = 2^4"},
{"2^3 + 2^3 = 4^2"},
{"4^3 + 4^3 = 2^7"},
{"3^3 + 6^3 = 3^5"},
{"4^3 + 8^3 = 24^2"},
{"8^3 + 8^3 = 2^10"},
{"8^3 + 8^3 = 4^5"},
{"8^3 + 8^3 = 32^2"},
{"9^3 + 18^3 = 3^8"},
{"9^3 + 18^3 = 9^4"},
{"9^3 + 18^3 = 81^2"},
{"16^3 + 16^3 = 2^13"},
{"7^3 + 21^3 = 98^2"},
{"18^3 + 18^3 = 108^2"},
{"22^3 + 26^3 = 168^2"},
{"16^3 + 32^3 = 192^2"},
{"11^3 + 37^3 = 228^2"},
{"32^3 + 32^3 = 2^16"},
{"32^3 + 32^3 = 4^8"},
{"32^3 + 32^3 = 16^4"},
{"32^3 + 32^3 = 256^2"},
{"2^3 + 46^3 = 312^2"},
{"25^3 + 50^3 = 375^2"},
{"27^3 + 54^3 = 3^11"},
{"50^3 + 50^3 = 500^2"},
{"10^3 + 65^3 = 525^2"},
{"14^3 + 70^3 = 588^2"},
{"36^3 + 72^3 = 648^2"},
{"56^3 + 65^3 = 671^2"},
{"64^3 + 64^3 = 2^19"},
{"28^3 + 84^3 = 28^4"},
{"28^3 + 84^3 = 784^2"},
{"33^3 + 88^3 = 847^2"},
{"72^3 + 72^3 = 864^2"},
{"65^3 + 91^3 = 1014^2"},
{"49^3 + 98^3 = 1029^2"},
{"65^3 + 104^3 = 1183^2"},
{"70^3 + 105^3 = 35^4"},
{"70^3 + 105^3 = 1225^2"},
{"57^3 + 112^3 = 1261^2"},
{"84^3 + 105^3 = 1323^2"},
{"88^3 + 104^3 = 1344^2"},
{"98^3 + 98^3 = 1372^2"},
{"64^3 + 128^3 = 1536^2"},
{"44^3 + 148^3 = 1824^2"},
{"128^3 + 128^3 = 2^22"},
{"128^3 + 128^3 = 4^11"},
{"128^3 + 128^3 = 2048^2"},
{"81^3 + 162^3 = 3^14"},
{"81^3 + 162^3 = 9^7"},
{"81^3 + 162^3 = 2187^2"},
{"8^3 + 184^3 = 2496^2"},
{"63^3 + 189^3 = 2646^2"},
{"96^3 + 192^3 = 24^5"},
{"114^3 + 190^3 = 2888^2"},
{"162^3 + 162^3 = 54^4"},
{"162^3 + 162^3 = 2916^2"},
{"100^3 + 200^3 = 3000^2"},
{"121^3 + 242^3 = 3993^2"},
{"200^3 + 200^3 = 4000^2"},
{"40^3 + 260^3 = 4200^2"},
{"65^3 + 260^3 = 65^4"},
{"65^3 + 260^3 = 4225^2"},
{"198^3 + 234^3 = 4536^2"},
{"78^3 + 273^3 = 4563^2"},
{"183^3 + 249^3 = 4644^2"},
{"56^3 + 280^3 = 4704^2"},
{"144^3 + 288^3 = 72^4"},
{"144^3 + 288^3 = 5184^2"},
{"242^3 + 242^3 = 5324^2"},
{"224^3 + 260^3 = 5368^2"},
{"256^3 + 256^3 = 2^25"},
{"256^3 + 256^3 = 32^5"},
{"99^3 + 333^3 = 6156^2"},
{"112^3 + 336^3 = 6272^2"},
{"154^3 + 330^3 = 6292^2"},
{"217^3 + 312^3 = 6371^2"},
{"169^3 + 338^3 = 6591^2"},
{"78^3 + 354^3 = 6696^2"},
{"132^3 + 352^3 = 6776^2"},
{"184^3 + 345^3 = 6877^2"},
{"288^3 + 288^3 = 6912^2"},
{"260^3 + 364^3 = 8112^2"},
{"37^3 + 407^3 = 8214^2"},
{"196^3 + 392^3 = 8232^2"},
{"273^3 + 364^3 = 91^4"},
{"273^3 + 364^3 = 8281^2"},
{"18^3 + 414^3 = 8424^2"},
{"330^3 + 345^3 = 8775^2"},
{"338^3 + 338^3 = 8788^2"},
{"329^3 + 371^3 = 9310^2"},
{"260^3 + 416^3 = 9464^2"},
{"34^3 + 450^3 = 9548^2"},
{"57^3 + 456^3 = 9747^2"},
{"242^3 + 433^3 = 9765^2"},
{"280^3 + 420^3 = 9800^2"},
{"228^3 + 448^3 = 10088^2"},
{"225^3 + 450^3 = 10125^2"},
{"336^3 + 420^3 = 10584^2"}
...
  1. n = 1000;
  2. asscubesum =
  3.   Table[i^3 + j^3 -> {i, j}, {i, n}, {j, i, n}] // Join @@ # & //
  4.    Last /@ Merge[#, List] &;
  5. asspower =
  6.   Table[i^j -> {i, j}, {i, 2, Sqrt[2 n^3]}, {j, 2,
  7.       Log[2 n^3]/Log[i]}] // Join @@ # & // Last /@ Merge[#, List] &;
  8. intersection = Intersection[Keys[asscubesum], Keys[asspower]];
  9. Tuples@{asscubesum[#], asspower[#]} & /@ intersection //
  10.     Join @@ Map[Flatten, #, {2}] & //
  11.    Map[ToString, #, {2}] & // #1 ~~ "^3 + " ~~ #2 ~~ "^3 = " ~~ #3 ~~
  12.       "^" ~~ #4 & @@@ # & // Column
复制代码

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回复 2# 青青子衿

这是什么...

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回复 3# Shiki
Wolfram Mathematica的代码,部分结果已经补上去了!

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回复 3# Shiki

简单来说就是用电脑软件来查找结果。

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回复 5# 爪机专用


    有没有不开挂的做法呢...

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