dioph4e

1.Introduction

Old days Euler expected that there was no integer solution in A⁴+B⁴+C⁴=D⁴.
Though it had been believed for a long time in the state, the following counterexample 
was discovered by Elkies in 1987.


       2682440⁴+15365639⁴+18796760⁴=20615673⁴ (1. Elkies)

After that, a smallest solution was discovered by Frye. (2.Frye).

　　　 95800⁴+217519⁴+414560⁴=422481⁴

3rd small solution was found by MacLeod  in 1997 in those days.

       630662624⁴+ 275156240⁴+ 219076465⁴=638523249⁴

Bernstein found some solutions as follows. (3.Bernstein).

       1390400⁴+2767624⁴+673865⁴=2813001⁴

       5507880⁴+8332208⁴+1705575⁴=8707481⁴

       5870000⁴+11289040⁴+8282543⁴=12197457⁴ 

       Seven solutions were found in the range of D<2.1*10^7.


1. Noam D Elkies: On A⁴+B⁴+C⁴=D⁴,Mathematics of Computation,Oct.1988
2. Roger E. Frye: Finding  95800⁴+217519⁴+414560⁴=422481⁴ on the Connection Machine
3. Daniel J. Bernstein: ENUMERATING SOLUTION TO p(a)+q(b)=r(c)+s(d)


I found  new solutions below by the method of Elkies.
582665296⁴+ 260052385⁴+ 186668000⁴=589845921⁴

169218021322170204480680305⁴+1507524066882038472584786800⁴+1288056982586427591062203384⁴
=1677479490238223823661446513⁴                                
                                
                                
22495595284040⁴+7592431981391⁴+27239791692640⁴=29999857938609⁴

2.How to solve the equation A⁴+B⁴+C⁴=D⁴
    
　　　　It is same as to find the rational solutions of r⁴+s⁴+t⁴=1
        m,n:integer
      　x,y,r,s,t:rational
      　r=x+y,s=x-y

        (2m²+n²)y²=-(6m²-8mn+3n²)x²-2(2m²-n²)x-2mn................ (1) 

        (2m²+n²)t²=4(2m²-n²)x²+8mnx+(n²-2m²)...................... (2)
        
　　　　1. Choose (m,n), and find the rational solution (x,y) of  (1).
　　　　2. Parametrize the rational solution (x,y).
　　　　3. Substitute the rational solution x to (2). (It becomes the elliptic curve.).

　　　　4. Find the rational solutions for elliptic curve.
　　　　5. Convert the rational solutions, and obtain (r,s,t).



       
3.Search results

1. Find  95800⁴+217519⁴+414560⁴=422481⁴

　　Frye has already found this solution,but his method is different from Elkies.
    So,I will find a solution  by the method of Elkies theoretically.
　　Elkies pointed out that it was found with (m,n)=(20,-9)
　　Substitute (m,n)=(20,-9) to (1),(2)
　　

　　　　　881y² = -4083x²-1438x+360........................... (3)

　　　　　881t² = 2876x²-1440x-719............................ (4)
    Find solution for (3),and obtain (x,y)=(49/318,23/106), and parametrize it
   

　　      x=1/318(43169*k²-657351-121578k)/(881k²+4083)....... (5)

          y=-1/106(20263*k²-93909+285806k)/(881k²+4083)....... (6)
   Substitute (5) to (4) 
   
        
         Y²=19435071440k⁴-5351620404k³+130338882000k²-194951575764k-357457601448.... (7)
         Y=t(140079k²+649197)
    Find rational solution for (7),and obtain k=-59/81,Y=2444043484/6561
   
    Substitute k=-59/81 to (5),(6),then x=-159380/422481,y=85060/140827
　

    r=x+y=-159380/422481+85060/140827=95800/422481
    
    s=x-y=-159380/422481-85060/140827=-414560/422481

    t(140079k²+649197)=2444043484/6561
   
    t=217519/422481

    (95800/422481)⁴+(-414560/422481)⁴+(217519/422481)⁴=1　

    Consequently
　　95800⁴+217519⁴+414560⁴=422481⁴　


2. 630662624⁴+ 275156240⁴+ 219076465⁴=638523249⁴
    MacLeod has already found this solution.
    I will find it by the method of Elkies in the same way as the previous solution.

    
    Let (m,n)=(8,-5)

　　153y²=-779x²-206x+80
    153t²=412x²-320x-103

    parametrize it and obtain

    x=(51k²-34k-5221)/(14(17k²+779))............................................ (8)
    y=(17k²+7558k-779)/(42(17k²+779))........................................... (9)

    Y=-31790X⁴+36941X³-56158X²+28849X+22030.................................... (10)
    X=(k+2)/7
    Find rational solution for (10) and obtain X=-3015/9707
    Substitute k=-3015/9707*7-2 to (8),(9) then x=-59251064/212841083,y=452909432/638523249
    

　　r=x+y=-59251064/212841083+452909432/638523249=275156240/638523249

    s=x-y=-59251064/212841083-452909432/638523249=-630662624/638523249

    t=219076465/638523249
    Consequently

    630662624⁴+ 275156240⁴+ 219076465⁴=638523249⁴


3. 582665296⁴+ 260052385⁴+ 186668000⁴=589845921⁴

　　I found this solution.
    New solution for (10) is X=18247/19530 and obtain it in the same way as the previous example.
  
　　

　　Let k=18247/19530*7-2 then x=-107537637/393230614,y=-842717681/1179691842

    r=x+y=-107537637/393230614-842717681/1179691842=-582665296/589845921
    s=x-y=-107537637/393230614+842717681/1179691842=260052385/589845921
    t=186668000/589845921
    Consequently

    582665296⁴+ 260052385⁴+ 186668000⁴=589845921⁴


4. 1677479490238223823661446513⁴=169218021322170204480680305⁴
                                +1507524066882038472584786800⁴
                                +1288056982586427591062203384⁴

    I found this solution too.
　　As a matter of fact, we can make big size solutions.
    Elkies proved that there were infinitely many rational solutions of (10).
    Now as solutions of (10),we know three points P(-31/467,30731278/467^2),
    Q(-3015/9707,438152930/5542697),R(18247/19530,292135420/3814209).
　　If we find the 4th point S which passes P, Q, R and meets the elliptic curve of (10),
    4th point S must be rational point. 
　　　　
　　Y=-12884004827215/50505407906X²+15872118910607/101010815812X
     +15400754666545/101010815812
    Find a point of intersection with the this curve and the (10),
    and obtain X=99656645595927/152926786974106

    k=99656645595927/152926786974106*7-2
    x=-446102015186622756034702165/1118319660158815882440964342
    y=1676742088204208677065467105/3354958980476447647322893026

    r=x+y=169218021322170204480680305/1677479490238223823661446513
    s=x-y=-1507524066882038472584786800/1677479490238223823661446513

    Consequently

    1677479490238223823661446513⁴=169218021322170204480680305⁴
                                 +1507524066882038472584786800⁴
                                 +1288056982586427591062203384⁴


5. 1670617271⁴+ 632671960⁴+ 50237800⁴=1679142729⁴

    I found this solution.
    Solve the simultaneous equation (1),(2) on (m,n)=(20,-9).
    
    Y²=-19956651632k⁴+49953492900k³+8463446784k²+566201542500k+91714580856
    Find rational solution and obtain k=-457/2836

    x=-1037945311/3358285458
    y=767763077/1119428486
    r=x+y=632671960/1679142729
    s=x-y=-1670617271/1679142729

    Consequently

　　1670617271⁴+ 632671960⁴+50237800⁴=1679142729⁴


6. 22495595284040⁴+7592431981391⁴+27239791692640⁴=29999857938609⁴

    I found this solution.
    Solve the simultaneous equation (1),(2) on (m,n)=(20,-9).

    Y²=-19956651632k⁴+49953492900k³+8463446784k²+566201542500*k+91714580856
    Find rational solution and obtain k=9319/12137

    x=-30088027265431/59999715877218

    y=4967721100883/19999905292406

    r=x+y=-7592431981391/29999857938609

    s=x-y=-22495595284040/29999857938609

    Consequently

      22495595284040⁴+7592431981391⁴+27239791692640⁴=29999857938609⁴


7. 3393603777⁴=3134081336⁴+2448718655⁴+664793200⁴

　　I found this solution.
    New solution for (10) is X=30671//229738 and obtain it in the same way as the previous example.
  　
　　Let k=30671/229738*7-2 then x=-1037837285/2262402518,y=-1783925455/6787207554
 
    r=x+y=-2448718655/3393603777

    s=x-y=-664793200/3393603777

    t=3134081336/3393603777

    Consequently

    3393603777⁴=3134081336⁴+2448718655⁴+664793200⁴

8. 5821981400⁴+15355831360⁴+140976551⁴=15434547801⁴

    I found this solution.
    Solve the simultaneous equation (1),(2) on (m,n)=(20,-9).

    Y²=19435071440k⁴-5351620404k³+130338882000k²-194951575764k-357457601448
    Find rational solution and obtain k=832289/426669


    x=-4766924980/15434547801
    y=-3529635460/5144849267

    r=x+y=-15355831360/15434547801

    s=x-y=5821981400/15434547801

    Consequently

      5821981400⁴+15355831360⁴+140976551⁴=15434547801⁴

9. 4987588419655⁴+2480452675600⁴+502038853976⁴=5062297699257⁴

　　I found this solution.
    New solution for (10) is X=14392467/14506151 and obtain it in the same way as the previous example.
  　
　　Let k=14392467/145061518*7-2 then x=-835711914685/3374865132838,y=7468041095255/10124595398514
 
    r=x+y=2480452675600/5062297699257

    s=x-y=-4987588419655/5062297699257

    t=502038853976

    Consequently

    4987588419655⁴+2480452675600⁴+502038853976⁴=5062297699257⁴
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