求两个整数,使得和为立方数,平方和也为立方数

wayne

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

1. {26,-18,2,10}
   
2. {286,-259,3,53}
   
3. {53586,31598,44,1570}
   
4. {14829166,-14602185,61,75661}
   
5. 
   

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northwolves

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}}

wayne

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.

nyy

用pq来表达mn，然后用pq穷举！

mathe

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
[X,Y]=
[6, -18]
[-3, -9]
[-2, 10]
[33/4, 207/8]
[366, -7002]
[109/25, -1727/125]
[-6474/1681, -491598/68921]
[-7887/8464, 8061993/778688]
[294838/25281, 165454370/4019679]
[13845597/151321, -51522644169/58863869]
然后从中反解x,y,u,v,p,m,n,q

mathe

p=u-v,q=u^2+v^2

wayne

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. {1,0,1,1}
   
2. {26, -18, 2, 10}
   
3. {286, -259, 3, 53}
   
4. {53586, 31598, 44, 1570}
   
5. {14829166,-14602185,61,75661}
   
6. {59638861334,-24673078334,3270,16090058}
   
7. {135494650136115,-48914984512196,44239,2747997481}
   
8. {11654837602635354174,-11541643128207161918,483736,6455672878210}
   
9. {234192173776567982667691,113516496202066695693956,70318863,4076285249165273}
   
10. {397899590410147126796502213942,-351607148900013328469170353694,3590624822,65572975496524741450}
    

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wayne

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 [294838/25281, 165454370/4019679] 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

wayne

OK ,我知道了,从{-3, -9}, {366, -7002}两个点可以产生 {-(6474/1681), 491598/68921}

wayne

终于用Mathematica梳理完了.

1. nextp[pp1_,pp2_]:=Module[{p1=pp1,p2=pp2,p},p=If[p1[[1]]==p2[[1]],(3p1[[1]]^2)/(2p1[[2]]),(p1[[2]]-p2[[2]])/(p1[[1]]-p2[[1]])];{p^2-p1[[1]]-p2[[1]],p(p^2-p1[[1]]-p2[[1]]-p1[[1]])+p1[[2]]}];
   
2. points=pts=SolveValues[y^2==x^3+108&&x^2<10^1,{x,y},Integers];
   
3. sol=Nest[SortBy[Union[Join[#,Table[nextp@@p,{p,Subsets[#,{2}]}]]],Numerator[Abs[#[[1]]]]&]&,points,3];
   
4. final=Table[{s,t={x,y,1}/GCD@@{x,y}/.First@Solve[{s[[1]]==6/(x-y),s[[2]]==(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[[1;;2]]]},{s,sol}];
   
5. SortBy[Select[final,#[[2,1]]>#[[2,2]]&],Numerator[Abs[#[[2,1]]]]&]//Column

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1. {{6,18},{1,0,1,1}}
   
2. {{-3,-9},{26,-18,2,10}}
   
3. {{-2,-10},{286,-259,3,53}}
   
4. {{33/4,207/8},{53586,31598,44,1570}}
   
5. {{366,-7002},{14829166,-14602185,61,75661}}
   
6. {{109/25,1727/125},{59638861334,-24673078334,3270,16090058}}
   
7. {{-(6474/1681),-(491598/68921)},{135494650136115,-48914984512196,44239,2747997481}}
   
8. {{-(7887/8464),-(8061993/778688)},{11654837602635354174,-11541643128207161918,483736,6455672878210}}
   
9. {{294838/25281,-(165454370/4019679)},{234192173776567982667691,113516496202066695693956,70318863,4076285249165273}}
   
10. {{13845597/151321,-(51522644169/58863869)},{397899590410147126796502213942,-351607148900013328469170353694,3590624822,65572975496524741450}}
    
11. {{248589006/81739681,8622316786662/739008455921},{195006235510562634327164261983395030,-142447923354321390494513663190454609,374582200541,387795679161298258653661}}
    
12. {{-(1330341839/298252900),-(22603078860841/5150827583000)},{2452444027749634442584550081702851471509358,167068701402841803280026345411099894722642,137850021357180,18213902411765061540558454562}}
    
13. {{18305505726/119079516241,427046113917344538/41091840384928039},{26707292140601968931822009472748990934730742036230,-26706125204606815661578142039298894283553889339851,1052807601737059,1125702363560673081950265303881701}}
    
14. {{21856907010237/1253842302001,-(103220473208752736631/1403991171507921751)},{4646426566234893025313824972357597533730271953123793738298,-302757665291658174981968538428129864099730829293000927986,16316195654413260658,278842766453645183103844901466781599370}}
    
15. {{866049860562118/21301187933041,-(25507213200362392061950/98311819992310721161)},{618884503937011846597152234565236228737477059147342076054992397369,-403354269393122271431414507722939325370159672074239698036809020856,5995647162749122514817,81718681821075803389786259263307634253836113}}

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