x/y+y/z+z/x=n的整数解

wayne

求$x/y+y/z+z/x=n$的整数解，要求$gcd(x,y,z)=1$.

nyy

1,1,1,3
1,2,4,5

lsr314

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

hujunhua

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`

hujunhua

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。

mathe

三次曲线，应该是椭圆函数。
X^2+XY^2+Y-nXY=0

hujunhua

{{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}}

hujunhua

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}}

mathe

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

wayne

对于给定的n,可否给出对应的所有解。