求两个整数,使得和为立方数,平方和也为立方数
知乎的一个帖子: https://www.zhihu.com/question/1898114608683787203也就是对于整数$m,n,$存在$ m+n=p^3, m^2+n^2=q^3$, 其中$p,q$为整数,且不为0.
因为可以从$(m,n,p,q) \to (d^3m,d^3n, dp, d^2q)$,所以$(d^3m,d^3n, dp, d^2q)$这种是平凡解,我们只讨论(m,n,p,q) 非平凡解.
我搜到的有4组
{26,-18,2,10}
{286,-259,3,53}
{53586,31598,44,1570}
{14829166,-14602185,61,75661}
试着列了几个解，似乎也没什么规律
{{26,-18,2,10},{286,-259,3,53},{208,-144,4,40},{702,-486,6,90},{2288,-2072,6,212},{1664,-1152,8,160},{7722,-6993,9,477},{3250,-2250,10,250},{5616,-3888,12,360},{18304,-16576,12,848},{8918,-6174,14,490},{13312,-9216,16,640},{18954,-13122,18,810}} northwolves 发表于 2025-4-25 17:14
试着列了几个解，似乎也没什么规律
{{26,-18,2,10},{286,-259,3,53},{208,-144,4,40},{702,-486,6,90},{22 ...
还需要根据$(m,n,p,q) \to (d^3m,d^3n, dp, d^2q)$ 来过滤掉,去重. 我是算到了$0<q<10^7$,只有4组非平凡解. 用pq来表达mn，然后用pq穷举！ 应该可以用椭圆曲线描述。
我们可以设\(m+ni=(u+vi)^3\),
得到\(m=u^3-3uv^2,n=3u^2v-v^3\),代入第一式，得到关于u,v,p齐次三次方程，可以转化为椭圆曲线。
设\(x=\frac up, y=\frac vp\)得到\((x-y)^3+6(x-y)xy=1\)
设\(X=\frac6{x-y},Y=\frac{18(x+y)}{x-y}\)
得到\(Y^2=X^3+108\)
通过Pari/gp可以产生一系列点如
=
[-3, -9]
[-2, 10]
[-6474/1681, -491598/68921]
[-7887/8464, 8061993/778688]
然后从中反解x,y,u,v,p,m,n,q p=u-v,q=u^2+v^2 mathe 发表于 2025-4-25 18:49
应该可以用椭圆曲线描述。
我们可以设\(m+ni=(u+vi)^3\),
得到\(m=u^3-3uv^2,n=3u^2v-v^3\),代入第一式，得 ...
转化一下,就是
{1,0,1,1}
{26, -18, 2, 10}
{286, -259, 3, 53}
{53586, 31598, 44, 1570}
{14829166,-14602185,61,75661}
{59638861334,-24673078334,3270,16090058}
{135494650136115,-48914984512196,44239,2747997481}
{11654837602635354174,-11541643128207161918,483736,6455672878210}
{234192173776567982667691,113516496202066695693956,70318863,4076285249165273}
{397899590410147126796502213942,-351607148900013328469170353694,3590624822,65572975496524741450}
mathe 发表于 2025-4-25 18:49
应该可以用椭圆曲线描述。
我们可以设\(m+ni=(u+vi)^3\),
得到\(m=u^3-3uv^2,n=3u^2v-v^3\),代入第一式，得 ...
[-6474/1681, -491598/68921] 跟 这个有理解是怎么生成的.好像不是切线产生的. PARI/Gp不会玩.Mathematica几乎没有椭圆曲线相关的支持 OK ,我知道了,从{-3, -9}, {366, -7002}两个点可以产生 {-(6474/1681), 491598/68921} 终于用Mathematica梳理完了.
nextp:=Module[{p1=pp1,p2=pp2,p},p=If]==p2[],(3p1[]^2)/(2p1[]),(p1[]-p2[])/(p1[]-p2[])];{p^2-p1[]-p2[],p(p^2-p1[]-p2[]-p1[])+p1[]}];
points=pts=SolveValues;
sol=Nest}]]],Numerator]]]&]&,points,3];
final=Table[{s,t={x,y,1}/GCD@@{x,y}/.First@Solve[{s[]==6/(x-y),s[]==(18(x+y))/(x-y)},{x,y}];{u^3-3 u v^2,3 u^2 v-v^3,((u-v) (u^2+4 u v+v^2))^(1/3),u^2+v^2}/.Thread[{u,v}->t[]]},{s,sol}];
SortBy]>#[]&],Numerator]]]&]//Column
{{6,18},{1,0,1,1}}
{{-3,-9},{26,-18,2,10}}
{{-2,-10},{286,-259,3,53}}
{{33/4,207/8},{53586,31598,44,1570}}
{{366,-7002},{14829166,-14602185,61,75661}}
{{109/25,1727/125},{59638861334,-24673078334,3270,16090058}}
{{-(6474/1681),-(491598/68921)},{135494650136115,-48914984512196,44239,2747997481}}
{{-(7887/8464),-(8061993/778688)},{11654837602635354174,-11541643128207161918,483736,6455672878210}}
{{294838/25281,-(165454370/4019679)},{234192173776567982667691,113516496202066695693956,70318863,4076285249165273}}
{{13845597/151321,-(51522644169/58863869)},{397899590410147126796502213942,-351607148900013328469170353694,3590624822,65572975496524741450}}
{{248589006/81739681,8622316786662/739008455921},{195006235510562634327164261983395030,-142447923354321390494513663190454609,374582200541,387795679161298258653661}}
{{-(1330341839/298252900),-(22603078860841/5150827583000)},{2452444027749634442584550081702851471509358,167068701402841803280026345411099894722642,137850021357180,18213902411765061540558454562}}
{{18305505726/119079516241,427046113917344538/41091840384928039},{26707292140601968931822009472748990934730742036230,-26706125204606815661578142039298894283553889339851,1052807601737059,1125702363560673081950265303881701}}
{{21856907010237/1253842302001,-(103220473208752736631/1403991171507921751)},{4646426566234893025313824972357597533730271953123793738298,-302757665291658174981968538428129864099730829293000927986,16316195654413260658,278842766453645183103844901466781599370}}
{{866049860562118/21301187933041,-(25507213200362392061950/98311819992310721161)},{618884503937011846597152234565236228737477059147342076054992397369,-403354269393122271431414507722939325370159672074239698036809020856,5995647162749122514817,81718681821075803389786259263307634253836113}}
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