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楼主 |
发表于 2025-6-24 19:59:26
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应该不算挖坟吧,关于$35x^2+y^3=z^3$ , 我再继续 往前推进一下. 因为还有价值.
首先, 不难得证, $5|z-y$, 所以 设$z=y+5k$, 得到$7 x^2 =25 k^3+15 k^2 y+3 k y^2$,
这个三参数的方程就比较有意思了, 对于给定的$k$, 是关于$x,y$的二次方程. 所以要么无解,要么存在 有理参数解.
1) 当$k=3m$, 即$z-y=5k=15m$的时候,$m$只能是形如$x^2 + xy + y^2$的数,叫做洛希数(Löschian number).其他无解. https://oeis.org/A003136, https://oeis.org/A270672
2) 当$k=3m+1$,即$z-y=5k=15m+5$ 的时候,$m$只能是形如$x^2 + xy + y^2 + x+ y$的数, 其他无解https://oeis.org/A202822.
3) 当$k=3m+2$的时候,无解.
以下Mathematica生成所有有解的k.
- nf[{i_,j_}]:=3(i^2+i*j+j^2+i+j)+1;Union[Flatten[{Select[Range[1,600],Resolve[Exists[{x,y},Reduce[#==3 (x^2+x y+y^2),{x,y},Integers]]]&],Union[nf/@Tuples[Range[-10,10],2]]}]]
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- 1,3,4,7,9,12,13,16,19,21,25,27,28,31,36,37,39,43,48,49,52,57,61,63,64,67,73,75,76,79,81,84,91,93,97,100,103,108,109,111,112,117,121,124,127,129,133,139,144,147,148,151,156,157,163,169,171,172,175,181,183,189,192,193,196,199,201,208,211,217,219,223,225,228,229,237,241,243,244,247,252,256,259,268,271,273,277,279,283,289,291,292,300,301,304,307,309,313,316,324,325,327,331,333,336,337,343,349,351,361,363,364,367,372,373,379,381,387,388,397,399,400,403,409,412,417,421,427,432,433,436,439,441,444,448,453,457,463,468,469,471,475,481,484,487,489,496,499,507,508,511,513,516,523,525,529,532,541,543,547,549,553,556,559,567,571,576,577,579,588,589,592,597,601,603,604,607,613,619,624,625,628,631,633,637,643,651,652,657,661,669,673,675,676,679,684,687,688,691,700,703,709,711,721,723,724,727,729,732,733,739,741,751,756,757,763,768,769,772,775,777,784,787,793,796,804,811,813,817,819,823,829,831,832,837,841,844,847,849,853,859,867,868,871,873,876,877,883,889,892,900,903,907,912,916,919,921,925,927,931,937,939,948,949,961,964,967,972,973,975,976,981,988,991,993,997,999,1009
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------
拿到了有解的k, 接下来就好说了,比如 $k=4, -1600 + 7 x^2 - 240 y - 12 y^2=0$的有理参数解是$[x,y]=[(-10*U^2 - 120*U - 840)/(U^2 - 84), (-5*U^2 + 140*U + 1260)/(U^2 - 84)]$
进而得到$35x^2+y^3=z^3$的有理参数解是 $[x,y,z]= [-\frac{10 \left(U^2+12 U+84\right)}{U^2-84},-\frac{5 \left(U^2-28 U-252\right)}{U^2-84},\frac{5 \left(3 U^2+28 U-84\right)}{U^2-84}]$
拿到 有理解的参数表达式了, 就可以 随便造了
- SortBy[Table[{tmp={(10 (84+12 U+U^2))/(-84+U^2),-((5 (-252-28 U+U^2))/(-84+U^2)),(5 (-84+28 U+3 U^2))/(-84+U^2)};d=Denominator[tmp[[1]]];d^3tmp[[1]],d^2tmp[[2]],d^2tmp[[3]]},{U,-10,10,1/10}],Abs]
复制代码- {-8,-12,8}
- {-10,-5,15}
- {-10,-15,5}
- {-14,-1,19}
- {-14,-19,1}
- {-26,9,29}
- {-40,20,40}
- {-40,-40,-20}
- {40,-40,-20}
- {-62,-57,-37}
- {-122,-103,-83}
- {-190,135,155}
- {190,135,155}
- {-190,-155,-135}
- {-910,-705,-685}
- {910,-705,-685}
- {-950,-265,235}
- {-2150,-545,-45}
- {4360,3320,3340}
- {12050,-2085,-1585}
- {-37570,-2635,3145}
- {-37570,-3145,2635}
- {-38726,-2397,3383}
- {46850,-7405,-6905}
- {57650,8555,9055}
- {-80920,-6120,-340}
- {-106930,-7395,-1615}
- {-121958,-8109,-2329}
- {-163574,4267,10047}
- {224450,34035,34535}
- {-367030,13515,19295}
- {367030,-19295,-13515}
- {-383320,-13320,14060}
- {-413438,-16909,10471}
- {-521110,-16605,17015}
- {-676286,-25197,2183}
- {-684130,-12685,24295}
- {-684130,-24295,12685}
- {-713714,-11653,25327}
- {-773942,-37621,-31841}
- {-799658,-24581,19599}
- {-870758,-3813,29807}
- {-917230,3515,30895}
- {-949870,-13395,30785}
- {-961480,-31820,5160}
- {-999370,-32745,-5365}
- {-1327990,5945,39565}
- {-1327990,-39565,-5945}
- {-1647914,-45637,-1457}
- {-1705690,-25665,43955}
- {1754230,75905,81685}
- {-1754230,-81685,-75905}
- {-2378470,103955,109735}
- {-2459170,-60845,-23865}
- {-2645560,-7080,62540}
- {-2666258,-64629,-27649}
- {-2810114,-63717,26063}
- {-3241058,-71221,18559}
- {-3600670,-79195,-35015}
- {-3775526,-86469,-52849}
- {-3807010,-54905,69915}
- {-3936902,55883,89503}
- {4123270,-90945,-53965}
- {-4202830,-100085,-72705}
- {4613530,-108595,-81215}
- {-4623670,1005,90785}
- {-4793530,54755,98935}
- {-5028970,-92545,45235}
- {-5866690,92045,125665}
- {-5882890,-108265,-38645}
- {-6019960,-55180,103240}
- {-6118840,-130380,-96760}
- {-6343550,-33405,111095}
- {-6384326,-109381,49039}
- {-6682330,-115785,21995}
- {-7047730,31155,120935}
- {-7905158,-129317,29103}
- {-8058790,-86355,117665}
- {-10811710,-86655,150965}
- {-10811710,-150965,86655}
- {-11296210,-165505,-40685}
- {-11395270,-514845,-509065}
- {-11522110,-182605,-112985}
- {11522110,-182605,-112985}
- {-16514590,-212005,25615}
- {-16612870,-125095,197485}
- {17053480,-298920,-254740}
- {-18783494,-366581,-332961}
- {19654870,330455,367435}
- {-20784070,-280395,-190615}
- {-21279640,-246280,96940}
- {-21465910,-257655,-99235}
- {22068280,441780,469160}
- {-22818970,-392685,-348505}
- {-23853790,-269205,74015}
- {-24890440,-280780,-76760}
- {-25764350,-299965,-155465}
- {-28055890,-539355,-505735}
- {-28273240,-283880,172140}
- {29266210,-561905,-528285}
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