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}} 这个问题挺有意思的。我现在才关注到。
https://mp.weixin.qq.com/s/S7563ci1lvd4d-r4KDZXjA
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, ,
发现解答过程 完全就是短文的英译中.
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/The%20Mathematical%20Gazette%20%201952-05_%20Vol%2036%20Iss%20316%20--%20Volume%2036,%20Issue%20316,%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。
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