A^4 + B^4 + C^4 = D^4的正整数解

wayne

-136/133,  $U^3 + 190637722732426876976/3*U + 3095889549021797083748946222464/27 + T^2$, 秩为2-4, PARI/GP 返回的生成元不稳定.

1. {16003017,14173720,12552200,4479031}
   
2. {156646737,146627384,108644015,27450160}
   
3. {409840652625395469143913,381461080909525552802665,281048473715879152495040,168213921178037201816584}
   
4. {174088703841632292189275073,170289556324371670328363560,92419682545114696981174360,46402888024739111034420161}
   
5. {2238064411114509557621828988264199107447781726361823098673,2113369080002372774659883075597987274223419109671305965361,1492990442221220647777007749959299887421662491298250867240,644720593112148423288344119361118447575892058103987575240}
   
6. {10334039288571423369814874726646837894863672090837769254061897,9726652415499527317841034071891584230243922225857544471903840,7038002871877350277682203335010157095362865491218618411066185,802222541842075087973421141552698482273947854957486056736504}
   
7. {66990775244825626736335667455603229442781393715909689864262553,63532363642825437016235185257645548111135632895531949789851545,43477481473417054643292362944820406498190722928604512949684104,22891794339552942427184246044914531340816245292069626082996000}
   
8. {6290753782969063233661330434356534242376804882255799075738904737,6004714485223259806236460566262042597527552068779040160357265240,3756515702750654460713731151780155211301221951849500370046000991,2859493431285398069961716836262941392834906534583762706237613160}
   
9. {5528044062633492498989146072933876479303792632980437783199462877094016466073,5146101434862667390289884519765799171337209166396804584956013489142187947161,3905026864373816766273646595601470257097303134741996523413579704662374185320,347018378881179480776386640093089345808622467628686725013220696033423765400}
   
10. {576997229245647003060694381078495433938663462789033420859793468243072975618780327393,550735739483116591547774395995830749789334008158848186623961903140964282561976366776,318753900464500655005013397741120816529879169295871488895424791130378589215294477855,303809589589865846563983170274914725587778302788848794227600745272610014043704992400}
    
11. {320104068499255317700941044099064410653243346293339550693110969912882986891170068607417165909648021118593553,278325081341790705124989870706565027560046245291650569735770015931675650161895952455253219329167760006512560,258947763769080106360551526007457444746748608819706365051324636297318283185334617707732140181027759952373265,39246729767295583013249868026312094474880502231228481707423947998908880577744254564214944794036637462332104}
    

复制代码

-56/165, $V^3 + T^2 + 32162600536911467536*V + 210821564607026400956984402048$, 秩为2-3

1. {1367141947873,1226022682752,1047978087905,408600530760}
   
2. {228746036963039501833,210878774189729581880,163180699054891578792,84616109521023161865}
   
3. {10739931407728904606857,10320518856970101984393,6653143628547990852040,772654695228940017240}
   
4. {50850387717526082374854170580381669126913882753,47994627294635220655319628420215891231758648520,34275156842649042025698564180116719276001302128,1196165497178991078511743748654292211747408255}
   
5. {100170624011022249554316482261662969143981102694273,99943854383556748642437114426169740886814750129025,26089512383088709960572508724554492604323228123960,25832835605608106736294184324057548118430642133648}
   
6. {13656184903701761046168458970860397901756208186306737114064425211995751450283582074416458697,13609595575212360971043002087200354959867954330982498494227314624823082574364638611644501160,4334661413525976887153352784109745981280977103912657569460129986083143588438562261822705320,3303852624768162799356646007436144536659101345640485583634225498450823475589432440955067447}
   
7. {1025129545560231132103805992080195646777288314811086732638222150995133156866584665849992451977033,1023294756546166338722567047625551431208316665757173162880744606498548985028492066124880268699465,296525411634209841178203319314620004809565320138716622675560045049547763450045991061803285016408,111407850305113530734592925331133923151946295814063878818522477318674842640315756383934485631400}
   
8. 
   

复制代码

xiaoshuchong

wayne 发表于 2025-6-7 18:53
-136/133,  $U^3 + 190637722732426876976/3*U + 3095889549021797083748946222464/27 + T^2$, 秩为2-4, PA ...

第二条可以确认是rank 3，ellrank可以给出前两个生成元，第三个可以用magma算
e = ellinit([2010162533556966721, -3294086946984787514952881282])
高度分别是 10.40  64.45 127.40
[1203084369, 29422218266224];
[1889568329531809891184337111758689/1503131618622411238293504, 64353005383546607819561727075447827880420052992145/1842873459592804748883596128454443008];
[4493857499019323354117464345068131100516769527262707199035081/1230765370930958542070668193165781965011255022089744 , -9914366570276210582765942740861455097373564986829083529537183532058367239905959710656256517/43178037832532557417215606108293843330503108215267371734186121190810066494528];

直接用magma的Generators得不到结果，因为有很多覆盖需要算，120秒不够。
以下是4-descent代码：
SetSeed(1);
SetClassGroupBounds("GRH");
E := EllipticCurve([2010162533556966721, -3294086946984787514952881282]);
P1:= E![1203084369, 29422218266224,1];
P2:= E![2316653632615804750911870180321143132626432128, 64353005383546607819561727075447827880420052992145, 1842873459592804748883596128454443008];
twocovers := TwoDescent(E : RemoveTorsion := true, RemoveGens := {P1,P2}); twocovers;
fourcovers := FourDescent(twocovers[1] : RemoveTorsion := true, RemoveGensEC := {P1,P2}); fourcovers;
m := 1;
pts := PointSearch(fourcovers[m],10^6:OnlyOne:=true);
pts;
_, maps4toE := AssociatedEllipticCurve(fourcovers[m] : E := E);
maps4toE(pts[1]);

wayne

PARI/Gp的代码

1. (250725.18:23:47)> E=ellinit([32162600536911467536,-210821564607026400956984402048])
   
2. %1 = [0, 0, 0, 32162600536911467536, -210821564607026400956984402048, 0, 64325201073822935072, -843286258428105603827937608192, -1034432873296937819820631171661181911296, -1543804825771750441728, 182149831820470810426834523369472, -21329839550959523878875172090160084446953355882212957132685312, 3508950721580750150835088745022049688200422668246890091952/20341720152816318396449253168258747527078014261448819287, Vecsmall([1]), [Vecsmall([128, -1])], [0, 0, 0, 0, 0, 0, 0, 0]]
   
3. 
   
4. (250725.18:24:15)> ellrank(E,10)
   
5. %3 = [3, 3, 2, [[4812337476, 235377746129792], [44965386980527188839236889364828676/408803660580707740551025, 9546841079275498607259404848820840423353036776935424/261379885854026647268706820725823625]]]
   
6. 
   
7. (250725.18:24:23)> ellrank(E,10)
   
8. %4 = [3, 3, 2, [[29110583830161/196, 157173307847093512633/2744], [44965386980527188839236889364828676/408803660580707740551025, 9546841079275498607259404848820840423353036776935424/261379885854026647268706820725823625]]]
   
9. 
   
10. (250725.18:25:55)> ellrank(E,10)
    
11. %5 = [3, 3, 2, [[4812337476, 235377746129792], [1889568329531809891184337111758689/375782904655602809573376, 64353005383546607819561727075447827880420052992145/230359182449100593610449516056805376]]]
    

复制代码

nyy

得到的方程的解呢？？

majer

分享一个相关的，去年日本一位数学爱好者かいもちゐ𓆡为带非0参数t的不定方程A⁴+B⁴+C⁴+D⁴+E⁴+tF⁴=G⁴，花9.5个小时找到的一组 解系。有时间的也可以验证一下

wayne

四个数都没完全整明白. 这都7个数了.

northwolves

太强大了，太变态了，太无聊了

wayne

用大模型提取了一下
\[\begin{align*}
A &= 111320336906015744973885469106473968722u^8 + 4464227773044791208628000u^4v^4t + 18818000000v^8t^2 \\
B &= 302361389866639583507334002929345728722u^8 + 4379637011392358328628000u^4v^4t + 18818000000v^8t^2 \\
C &= 765445922800453643969205027248473633083u^8 + 10205839102340109972942000u^4v^4t + 28227000000v^8t^2 \\
D &= 1164744593185150801810062993124575001444u^8 + 12043331989924890985256000u^4v^4t + 37636000000v^8t^2 \\
E &= 1478515310376600713196217637325867913444u^8 + 13888333930735762329256000u^4v^4t + 37636000000v^8t^2 \\
F &= 173795678072144627997205408331280000u^7v + 101916580304136000000u^3v^5t \\
G &= 1624881865027322098754362048979764321805u^8 + 16044921543687695821570000u^4v^4t + 47045000000v^8t^2
\end{align*}\]

然而Mathematica验证好像没通过.

1. Factor[a^4+b^4+c^4+d^4+e^4+t f^4-g^4/.{a -> 111320336906015744973885469106473968722u^8 + 4464227773044791208628000u^4v^4t + 18818000000v^8t^2 ,
   
2. b -> 302361389866639583507334002929345728722u^8 + 4379637011392358328628000u^4v^4t + 18818000000v^8t^2 ,
   
3. c ->765445922800453643969205027248473633083u^8 + 10205839102340109972942000u^4v^4t + 28227000000v^8t^2 ,
   
4. d -> 1164744593185150801810062993124575001444u^8 + 12043331989924890985256000u^4v^4t + 37636000000v^8t^2 ,
   
5. e-> 1478515310376600713196217637325867913444u^8 + 13888333930735762329256000u^4v^4t + 37636000000v^8t^2 ,
   
6. f -> 173795678072144627997205408331280000u^7v + 101916580304136000000u^3v^5t ,
   
7. g -> 1624881865027322098754362048979764321805u^8 + 16044921543687695821570000u^4v^4t + 47045000000v^8t^2}]

复制代码

northwolves

1. ClearAll["Global`*"];a=111320336906015744973885469106473968722u^8+4464227773044791208628000u^4 v^4 t+18818000000v^8t^2;
   
2. b=302361389866639583507334002929345728722u^8+4379637011392358328628000u^4 v^4 t+18818000000v^8t^2; c=765445922800453643969205027248473633083u^8+10205839102340109972942000u^4 v^4 t+28227000000v^8t^2;
   
3. d=1164744593185150801810062993124575001444u^8+12043331989924890985256000u^4 v^4 t+37636000000v^8t^2;
   
4. e=1478515310376600713196217637325867913444u^8+13883333930735762329256000u^4 v^4 t+37636000000v^8t^2;
   
5. f=173795678072144627997205408331280000u^7v+1019165803041360000000u^3v^5t;
   
6. g=1624881865027322098754362048979764321805u^8+16044921543687695821570000u^4 v^4 t+47045000000v^8t^2;a^4+b^4+c^4+d^4+e^4+t f^4-g^4//FullSimplify

复制代码

0