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[讨论] x/y+y/z+z/x=n的整数解

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发表于 2025-4-27 08:25:49 | 显示全部楼层 |阅读模式

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求$x/y+y/z+z/x=n$的整数解,要求$gcd(x,y,z)=1$.
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2025-4-27 10:42:23 | 显示全部楼层
1,1,1,3
1,2,4,5
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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发表于 2025-4-27 13:34:05 | 显示全部楼层
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
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2025-4-27 14:13:20 | 显示全部楼层

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`
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2025-4-27 14:58:08 | 显示全部楼层
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。

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2025-4-27 18:44:05 来自手机 | 显示全部楼层
三次曲线,应该是椭圆函数。
X^2+XY^2+Y-nXY=0
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2025-4-28 03:02:02 | 显示全部楼层

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}}
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2025-4-28 04:09:19 | 显示全部楼层

-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}}

毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2025-4-28 06:06:50 来自手机 | 显示全部楼层
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
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2025-4-28 23:32:43 来自手机 | 显示全部楼层
对于给定的n,可否给出对应的所有解。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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