求两个整数,使得和为立方数,平方和也为立方数

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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nyy

这个问题挺有意思的。我现在才关注到。

https://mp.weixin.qq.com/s/S7563ci1lvd4d-r4KDZXjA

wayne

nyy 发表于 2025-4-28 20:44
这个问题挺有意思的。我现在才关注到。

https://mp.weixin.qq.com/s/S7563ci1lvd4d-r4KDZXjA

这哥们提到的 1952年的短文:  Quartics with Values Which Are Perfect Squares
我下载下来了, 2289. Quartics with Values Which Are Perfect Squares (R. C. Lyness).pdf (369.44 KB, 下载次数: 2)

2025-4-28 22:19 上传

点击文件名下载附件

,
发现解答过程 完全就是短文的英译中.

wayne

这篇文章出自 数学公报 (The Mathematical Gazette), Volume 36 - Issue 316 - May 1952
这一期的杂志我也下载到了,  2289. Quartics with Values Which Are Perfect Squares。在 第53/73 页.
https://nestwhile.com/res/ebook/ ... 0316,%201952-05.pdf

《数学公报》是由剑桥大学出版社代表数学协会出版的三年一次的同行评审学术期刊。
它涵盖了以15至20岁年龄段为重点的数学教育。该杂志由爱德华·曼·兰利（Edward Mann Langley）于1894年创立，是《改进几何教学协会的报告》的继任者。