解方程：(x + y + z)(x^3 + y^3 + z^3) = n^6

northwolves

猜想：$(x + y + z) (x^3 + y^3 + z^3) =n^6$对于所有正整数n都有整数解。

比如：
{{1,0,0},1}
{{0,-2,-2},2}
{{3,3,3},3}
{{9,-8,-9},4}
{{15,-5,-5},5}
{{-51,-5,50},6}
{{-58,15,42},7}
{{-36,4,24},8}
{{-27,0,0},9}
{{-25,-15,-10},10}
{{-186,18,167},11}
{{-34,-30,16},12}
{{91,0,-104},13}
{{-70,0,42},14}
{{-131,23,99},15}
{{-164,24,132},16}
{{-119,51,51},17}
{{-213,50,157},18}
{{-1030,14,1015},19}
{{-408,-28,404},20}
{{-175,49,105},21}
{{-729,1,720},22}
{{},23}
{{-408,-40,400},24}
{{-125,0,0},25}
{{-624,27,589},26}
{{-81,-81,-81},27}
{{-464,120,336},28}
{{-240,70,110},30}
{{-341,93,217},31}
{{-1544,11,1525},32}
{{-781,-11,759},33}
{{-2079,7,2064},34}
{{-441,28,364},35}
{{-608,-4,564},36}
{{},37}
{{-608,-38,570},38}
{{-10283,-4927,10647},39}
{{-540,124,376},40}
{{},41}
{{-546,245,259},42}
{{},43}
{{-1488,144,1336},44}
{{-756,252,477},45}
{{},46}
{{-6627,6533,2303},47}
{{-272,-240,128},48}
{{0,0,-343},49}
{{-250,-250,0},50}
{{-1071,-119,1037},51}
{{-832,0,728},52}
{{},53}
{{},54}
{{-6050,5665,3410},55}
{{5292,-5824,3668},56}
{{},57}
{{},58}
{{},59}
{{39590,-39600,3610},60}
{{},61}
{{},62}
{{-1566,405,1134},63}
{{0,0,-512},64}
{{-7215,5915,5525},65}
{{-1035,-80,899},66}
{{-1206,469,670},67}
{{-952,408,408},68}
{{-1357,322,966},69}
{{-3630,-70,3600},70}

northwolves

注意到  $\{x,y,z,t\}=\{-6 k-\frac{6}{k}-3,6 k+\frac{6}{k+1}+3,\frac{6 k}{k+1}+\frac{6}{k}+3,3\}$ 是一组解
令$k=\{1,2}$,解得： {-15, 12, 12, 3}, {-18, 17, 10, 3}

mathe

对于给定的n,我们可以知道\(x+y+z|n^6\),
于是我们可以枚举\(n^6\)的因子d,让\(x+y+z=d\),
然后得到一个关于x,y的三次曲线\(-3x^2y-3xy^2 + 3dx^2 + 6dxy + 3dy^2 - 3d^2x  - 3d^2y =\frac{n^6}d-d^3\).
可以利用椭圆曲线求解工具求其所有整数点。

mathe

由于\(3(x+y)|\frac{n^6}d-d^3\), 可以继续通过因子分解来做，结果比想象的简单，而n=37无解。
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mathe

先提交一个无解序列
https://oeis.org/draft/A391526

wayne

mathe 发表于 2025-12-12 15:59
对于给定的n,我们可以知道\(x+y+z|n^6\),
于是我们可以枚举\(n^6\)的因子d,让\(x+y+z=d\),
然后得到一个关 ...

根据$x + y + z = d, x^3 + y^3 + z^3 = n^6/d$,消去z,得到 $3 (d - x) (d - y) (x + y) =\frac{d^4 - n^6}{d}$, 所以最好的算法可能还是 因子分解, 而不是椭圆曲线.

nindilg

mathe 发表于 2025-12-12 16:58
先提交一个无解序列
https://oeis.org/draft/A391526

oeis数列标记问题来源,fang的链接,
https://www.zhihu.com/question/1969018790831461543