x/y+y/z+z/x=n的整数解
求$x/y+y/z+z/x=n$的整数解，要求$gcd(x,y,z)=1$.1,1,1,3
1,2,4,5 齐次方程，可以令z=1,然后变成一个求椭圆曲线有理解的问题
由因子分析得等价方程
记$d_{xy}=gcd(x,y),d_{yz}=gcd(y,z),d_{zx}=gcd(z,x)$由于$gcd(x,y,z)=1$, 故而$d_{xy},d_{yz},d_{zx}$两两互质。
设$x=x'd_{xy}d_{zx},y=y'd_{xy}d_{yz},z=z'd_{zx}d_{yz}$
$1=1/d_{xy}gcd(x,y)=gcd(x'd_{zx},y'd_{yz})→gcd(x',y')=gcd(x',d_{yz})=gcd(y',d_{zx})=1$
$(x'd_{zx})/(y'd_{yz})+(y'd_{xy})/(z'd_{zx})+(z'd_{yz})/(x'd_{xy})=n$ → `x'^2z'd_{zx}^2d_{xy}+y'^2x'd_{xy}^2d_{yz}+z'^2y'd_{yz}^2d_{zx}=nx'y'z'd_{xy}d_{yz}d_{zx}`
检查上式两边的因子`x'`可知 `x'|z'^2y'd_{yz}^2d_{zx}→x'|d_{zx}`
检查上式两边的因子`d_{zx}`可知 `d_{zx}|y'^2x'd_{xy}^2d_{yz}→d_{zx}|x'`
所以`x'=d_{zx}`, 同理，`y'=d_{xy},z'=d_{yz}`, 代回可得
`x=x'^2y', y=y'^2z',z=z'^2x'`, 代入原方程得\[
\frac{x'^2}{y'z'}+\frac{y'^2}{z'x'}+\frac{z'^2}{x'y'}=n
\]化为整式, 去掉`'`号即得等价方程\[
x^3+y^3+z^3=nxyz\\
(x,y)=(y,z)=(z,x)=1
\]比如2#的两例即
`1^3+1^3+1^3=3·1·1·1\\
1^3+1^3+2^3=5·1·1·2` 容易再得一个小的解
`1^3+2^3+3^3=6·1·2·3`
不限于正数的话，还有一个小的解
`2^3+3^3+(-5)^3=3·2·3·(-5)`
由于 `x^3+y^3+z^3=nxyz→`\[ (x+y+z)(x+yω+zω^2)(x+yω^2+zω)=(n-3)xyz\tag1\]\[(ω^2+ω+1=0)
\]当n=3时，除了x=y=z=1, 不限于正数时还有`x+y+z=0`的任意解。 三次曲线，应该是椭圆函数。
X^2+XY^2+Y-nXY=0
1≤x≤y≤z≤1000以内的解, `x^3+y^3+z^3=nxyz`,（`x,y,z`两两互质）
{{1,1,1,3},
{1,1,2,5},
{1,2,3,6},
{1,2,9,41},
{1,3,14,66},
{1,5,9,19},
{1,5,14,41},
{1,9,146,2369},
{1,9,365,14803},
{1,14,45,149},
{1,14,61,269},
{1,14,549,21529},
{1,14,915,59802},
{1,35,54,106},
{1,35,794,18014},
{1,49,325,2163},
{1,49,362,2681},
{1,65,114,237},
{1,93,398,1725},
{1,99,626,3974},
{1,117,413,1491},
{1,147,962,6318},
{1,185,434,1097},
{1,234,329,629},
{2,3,7,9},
{2,5,133,1769},
{2,7,13,14},
{2,7,27,53},
{2,7,117,978},
{2,9,67,250},
{2,13,21,21},
{2,13,63,154},
{2,13,245,2309},
{2,13,735,20778},
{2,21,31,30},
{2,27,97,178},
{2,31,43,41},
{2,35,703,7061},
{2,43,57,54},
{2,57,73,69},
{2,73,91,86},
{2,91,111,105},
{2,91,657,2378},
{2,111,133,126},
{2,133,157,149},
{2,157,183,174},
{2,183,211,201},
{2,211,241,230},
{2,241,273,261},
{2,273,307,294},
{2,307,343,329},
{2,343,381,366},
{2,381,421,405},
{2,421,463,446},
{2,463,507,489},
{2,507,553,534},
{2,553,601,581},
{2,601,651,630},
{2,651,703,681},
{2,703,757,734},
{2,757,813,789},
{2,813,871,846},
{2,871,931,905},
{2,931,993,966},
{3,7,74,261},
{3,11,679,13971},
{3,14,163,633},
{3,22,305,1410},
{3,23,871,10995},
{3,43,494,1893},
{5,7,18,10},
{5,7,78,174},
{5,9,61,83},
{5,14,151,326},
{5,18,37,17},
{5,63,442,622},
{5,78,817,1713},
{5,254,481,209},
{7,11,279,1011},
{7,15,143,195},
{7,78,629,726},
{9,13,38,13},
{9,13,77,51},
{9,19,542,1718},
{9,38,91,26},
{9,49,377,323},
{9,77,409,243},
{9,611,790,166},
{11,38,259,161},
{13,23,378,478},
{13,42,95,18},
{15,19,731,1875},
{19,26,905,1658},
{19,91,310,57},
{19,746,945,94},
{23,31,567,451},
{27,43,182,29},
{27,182,673,94},
{42,95,523,69},
{70,151,629,38}}
-1000≤y≤x＜0＜z≤1000的解，`x^3+y^3+z^3=nxyz`，(`x,y,z`两两互质)
排除了n=3的解{{-1,-1,1,-1},
{-1,-2,9,40},
{-1,-3,7,15},
{-1,-3,28,261},
{-1,-4,13,41},
{-1,-4,65,1056},
{-1,-5,21,87},
{-1,-5,126,3175},
{-1,-6,31,159},
{-1,-6,217,7848},
{-1,-7,43,263},
{-1,-7,86,1056},
{-1,-7,172,4226},
{-1,-7,344,16905},
{-1,-8,57,405},
{-1,-8,513,32896},
{-1,-9,73,591},
{-1,-9,730,59211},
{-1,-10,91,827},
{-1,-11,111,1119},
{-1,-12,133,1473},
{-1,-13,157,1895},
{-1,-14,183,2391},
{-1,-14,305,6644},
{-1,-15,211,2967},
{-1,-16,241,3629},
{-1,-17,273,4383},
{-1,-18,307,5235},
{-1,-19,49,119},
{-1,-19,140,1029},
{-1,-19,343,6191},
{-1,-19,980,50547},
{-1,-20,381,7257},
{-1,-21,421,8439},
{-1,-22,463,9743},
{-1,-23,507,11175},
{-1,-24,553,12741},
{-1,-25,601,14447},
{-1,-26,81,244},
{-1,-26,217,1808},
{-1,-26,651,16299},
{-1,-27,37,31},
{-1,-27,532,10481},
{-1,-27,703,18303},
{-1,-28,757,20465},
{-1,-29,813,22791},
{-1,-30,871,25287},
{-1,-31,56,84},
{-1,-31,98,300},
{-1,-31,304,2978},
{-1,-31,532,9128},
{-1,-31,931,27959},
{-1,-32,993,30813},
{-1,-36,97,248},
{-1,-36,481,6424},
{-1,-38,201,1056},
{-1,-38,273,1956},
{-1,-62,273,1188},
{-1,-62,873,12288},
{-1,-65,146,299},
{-1,-122,779,4955},
{-1,-129,337,831},
{-1,-135,316,682},
{-1,-147,508,1713},
{-1,-161,277,383},
{-1,-182,417,876},
{-1,-182,711,2731},
{-1,-231,793,2655},
{-1,-254,527,971},
{-1,-259,380,381},
{-1,-259,988,3701},
{-1,-278,513,796},
{-1,-455,666,664},
{-1,-471,772,978},
{-2,-7,39,108},
{-2,-7,351,8800},
{-2,-9,737,30176},
{-2,-13,45,76},
{-2,-19,109,311},
{-2,-19,763,15320},
{-2,-73,819,4591},
{-3,-4,91,690},
{-3,-5,38,96},
{-3,-7,370,6519},
{-3,-8,539,12105},
{-3,-13,139,495},
{-3,-79,859,3111},
{-3,-112,643,1224},
{-3,-122,995,2700},
{-3,-247,763,759},
{-4,-5,189,1786},
{-4,-7,407,5916},
{-4,-9,61,103},
{-4,-9,793,17468},
{-4,-11,279,1769},
{-4,-21,373,1656},
{-4,-175,379,185},
{-5,-6,341,3876},
{-5,-7,52,77},
{-5,-7,117,391},
{-5,-9,854,16207},
{-5,-31,831,4455},
{-5,-67,252,186},
{-5,-99,434,376},
{-6,-7,559,7440},
{-6,-91,409,303},
{-7,-8,95,161},
{-7,-8,855,13054},
{-7,-9,67,71},
{-7,-9,268,1140},
{-7,-13,635,4431},
{-7,-18,247,484},
{-7,-20,927,6138},
{-7,-78,151,36},
{-7,-122,817,779},
{-7,-148,403,149},
{-7,-153,976,886},
{-8,-97,585,439},
{-9,-11,515,2679},
{-9,-13,133,151},
{-9,-17,434,1231},
{-9,-28,613,1491},
{-9,-31,70,16},
{-9,-38,329,316},
{-9,-103,208,41},
{-11,-13,882,5440},
{-12,-61,949,1230},
{-13,-14,61,20},
{-13,-14,549,1656},
{-13,-35,313,215},
{-13,-203,999,375},
{-13,-271,436,41},
{-14,-19,97,35},
{-18,-19,259,196},
{-19,-21,52,6},
{-19,-554,819,44},
{-28,-209,633,66},
{-38,-367,845,47},
{-39,-76,619,129},
{-45,-151,931,127},
{-91,-185,516,15},
{-151,-279,910,19}}
A085705
3, 5, 6, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 30, 31, 35, 36, 38, 40, 41, 44, 47, 51, 53, 54, 57, 62, 63, 64, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 83, 84, 86, 87, 92, 94, 96, 98, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 112, 113, 116
对于给定的n,可否给出对应的所有解。
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