1.Introduction

2010.4.11:  Add   New solutions 

Thank you  Mr.Jacobi.

He sent to me three solutions and one of them is a new solution.

Q=(k,Y) denotes the solution of (7).

Case 1. Q1=(46824943333 / 7233418750, -90519417593095489212844200389 / 46379919612500)

This solution gives a new solution(found by Lee Jacobi).

a = 4625798910,b = 46140636,c = 3744195265,d = -3172936050,e = 5243198761

Case 2. Q2=(1176017269462 / 440979965625, 109096878379234607793726950343916 / 172377467109253125)

This gives a 13 digit solution.

a=1058103081810, b=535945811334, c=-1140105961325, d=944080652640, e=1398023584459



I searched other solutions by using above two solutions Q1 and Q2.

First,convert the quartic solution to Weierstrass' one.

Next,generate some points of Weierstrass type curve by group law using p1 and p2.

p1=[862434907008475079082/898620529, -21281839968736771109964348820557/26937947597833]
p2=[286244140671159992513418/141706320721, 157502776841350834590549211749970749/53343785665892519]

Two new solutions were found where digit < 50.

27 digit solution: -p1+ p2=[1180574750827, -1240480222676161515]

a= 756005932676079581480132094
b= 254919457795946854721610090
c= 644145677060583739541816400
d= -718471658211690720060678365
e= 936599409320919455682880219

32 digit solution: p1-p2=[1180574750827, 1240479042101410688]

a= 47889177339639353814221283135714
b= -6535571193228450354843506097540
c= -26485662193954085869452902545110
d= 38071142693337417804472545034525
e= 52939086645794235394397419527589



2010.4.2:  Add   About transform quartic for to minimal Weierstrass form 

v^2=a*u^4+ b*u^3+ c*u^2+ d*u+ q^2--------> y^2+ a1*x*y+ a3*y = x^3+ a2*x^2+ a4*x +a6

First,we convert quartic to general Weierstrass form.

a1=d/q
a2=c-d^2/(4*q^2)
a3=2*q*b
a4=-4*q^2*a
a6=a*(d^2-4*q^2*c)

Substitute (a,b,c,d) to above convert form, we get
a1=129515154647285940/107
a2=-3946980368620254607138031941804008/11449
a3=354038064243767383338165499234355722500000
a4=-85551237005682176137274966636751507756798140625000000
a6=29493322820561289419204683782659854018385436191238340136057194420402433625000000000.

Next,we convert general form to standard reduced form.
I used PARI-GP(Package for number theory) as following.
I think PARI-GP is very convenient tool.
We can get a minimal form by using command "ellinit" and "ellminimalmodel".

e=ellinit([129515154647285940/107,-3946980368620254607138031941804008/11449,354038064243767383338165499234355722500000,-85551237005682176137274966636751507756798140625000000,29493322820561289419204683782659854018385436191238340136057194420402433625000000000]);
L1=ellminimalmodel(e,&v);

The vector v=[u,r,s,t] is outputted.
v=[96269732, -7178602271156816577346851192, -64757572173212308/107, 445927855213075033059760770402213392773170240/107]
v : Changing an old and new coodinate.
x=u^2*x1+r, y=u^3*y1+s*u^2*x1+t,(x1,y1) are the new coodinates.
First component of L1 is [1, -1, 0, 693712100835217413098595, -925623290959491513363159202180191099].
This is minimal Weierstrass form!





2009.7.30:  Add  4.Table of Solutions.

I knew in July that Jacobi and Madden[1] had already proved the infinity of the solution
for a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4. 

The feature of their methods is to use a known solution well. 

They used a=5400,b=1770,c=-2634,d=955 as a known solution.
 
I proved the infinity of the solution by using a known solution which is different from them,
and found some new solutions.

I used a=48150, b=-31764, c=27385, d=7590 as a known solution.

My method almost depends on their method.

[1]:Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4, Amer. Math. Monthly 115 (2008) 220-236. 



2. a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 has infinetly many solutions


  
Proof.

     Assume that a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4.

     We use identity a^4 + b^4 + (a + b)^4 = 2(a^2 + ab + b^2)^2.
   
     So, a^4 + b^4 + (a + b)^4 + c^4 + d^4 + (c +d )^4 = a + b)^4 + (c +d )^4 + (a+b+c+d)^4.

     (a^2 + ab + b^2)^2 + (c^2 + c*d + d^2)^2 = ( (a + b)^2 + (a + b)(c + d) + (c + d)^2 )^2

     We obtain 

     c^2 + cd + d^2 = m( (a + b)^2 + (a + b)(c + d) + (c + d)^2 -a^2 -ab -b^2) ..........(1)
     c^2 + cd + d^2 = 1/m( (a + b)^2 + (a + b)(c + d) + (c + d)^2 +a^2 +ab +b^2) ........(2)

     We use a known soulution to determine the value of m.

     48150^4 + (-31764)^4 + 27385^4 + 7590^4 = 51361^4 (Wroblewski)

     Set a0= 48150, b0= -31764, c0= 27385, d0= 7590 and substitute to (1).

     We get m=1807/475.

     To find the solution easy to obtain, we convert the variables.
 
     a = 2z + 2w
     b = 2z - 2w
     c = -x - y - z
     d = x - y - z

     (1),(2) become to follwing equation.

     -5979*z^2 + 1807*x^2 + 3521*y^2 + 10842*y*z - 1900*w^2 = 0 ................(3)

     1425*z^2 + 475*x^2 - 5803*y^2 + 2850*y*z + 7228*w^2 = 0 ...................(4)


     5415000*z^2 - 12158496*y^2 + 13963496*w^2 = 0 .............................(5)

     To find a parametric solution of (5),we use a known solution (y0,w0,z0) = (-21584, 39957/2, 8193/2).

     
     y = -2/2103*(256310000*k^2+575502144+97291875*k)...........................(6)

     z = 1/13319*(616181875*k^2+1383535524+14579387648*k)

     w = 225625*k^2 - 506604

     Substitute x,y,z of (6) to (3),then we get a quartic equation. 

     Y^2 = 65133101042606285634765625*k^4 + 9768720608022511028587687500*k^3
         + 21535806813463498836066039708*k^2 + 21934062869392293305824429200*k 
         + 328370811600539078296707600..........................................(7)

     Y = 72202299x


     Transform (7) to minimal Weierstrass form (8).

     V^2 + UV = U^3 - U^2 + 693712100835217413098595U - 925623290959491513363159202180191099...(8)
   
     We get a point P(U,V) = (434746275961695834/11449, -286717775104047608841677439/1225043)

     As this point on the curve (8) does not have intger coordinates,
     there are infinitely many rational points on the curve (8) by Nagell-Lutz theorem.

     Point 2P(U,V)=(5271161734852323519690712493513646148559824171/555979487782184246211680800056025,
              -383967827631115132325778408795184936425282259780975975668812642463671/13109561148524541584110100733586619386486747620125)

     k = 168657435152101942314450/1324296399347803844274341

     Y = -287049884891605933917141949569922910223374378499787229457540/4858063582619274122389752417972653863736484721

     x = -52951463732080046839539190106977109430616930178894526740/64704548856906112036109112454977776811106239998999

     Substitute k,x to (6),then we get

     y = -1917478095003366256203494101507639824416863283782973716/3405502571416111159795216444998830358479275789421

     z = 15790193338970706807327361229639263827375066043128677904/64704548856906112036109112454977776811106239998999

     w = -2443336111689201327548087382875038165845183854034984/4858063582619274122389752417972653863736484721

     We get a new solution.
     
     a = -99205298414251164662846807108945267922382794305394

     b = 286217560523954682289956279947366565354401483969850

     c = 217902012809334210448629156518650668768393853565040

     d = -95665174177424955524433769002923042240304835226765

     a+b+c+d = 309249100741612772551304860354148923960107708002731
      

     We can obtain infinitely many integer solutions of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 by apllying the group law.


    Q.E.D.



3. Small New Solutions

    (b + c + d)(a + c + d) = p*t^2
    (a + b + d)(a + b + c) = p*r^2/2
    a*b + a*c + a*d + b*c + b*d + c*d = -p*r*t .................................(9)

    Change the variables by (10).

    a =-2A + B + C + D
    b = A  -2B + C + D
    c = A  + B -2C + D
    d = A  + B + C -2D .........................................................(10)

    (9) becomes to (11).

    A*B = p*t^2
    C*D = p*r^2/2
    A*B + A*C + A*D + B*C + B*D + C*D - A^2 - B^2 - C^2 - D^2 = -3p*r*t ........(11)

    We use a known solution again.

    a0 = 48150, b0 = -31764, c0 = 27385, d0 = 7590

    A0 = b0 + c0 + d0 = 3211 = (13)^2*(19)

    B0 = a0 + c0 + d0 = 83125 = (5)^4*(7)*(19)

    C0 = a0 + b0 + d0 = 23976 = (2)^3*(3)^4*(37)

    D0 = a0 + b0 + c0 = 43771 = (7)*(13)^2*(37)

    Set

    A = p1*t1*t2^2
    B = p2*t1*t3^2
    C = p1*r1*r3^2/2
    D = p2*r1*r2^2 .............................................................(12)

    Substitute (12) to (11),then (11) becomes to (13).

    We obtain the equation similar to their lemma4.

    4*p1*t1^2*t2^2*p2*t3^2 + 2*p1^2*t1*t2^2*r1*r3^2 + 4*p1*t1*t2^2*p2*r1*r2^2 + 2*p2*t1*t3^2*p1*r1*r3^2
    +4*p2^2*t1*t3^2*r1*r2^2 + 2*p1*r1^2*r3^2*p2*r2^2 - 4*p1^2*t1^2*t2^4 - 4*p2^2*t1^2*t3^4
    -p1^2*r1^2*r3^4 - 4*p2^2*r1^2*r2^4 + 12*p1*p2*r1*r2*r3*t1*t2*t3 = 0 ............(13)


    a = -4*p1*t1*t2^2+2*p2*t1*t3^2+p1*r1*r3^2+2*p2*r1*r2^2

    b = 2*p1*t1*t2^2-4*p2*t1*t3^2+p1*r1*r3^2+2*p2*r1*r2^2

    c = 2*p1*t1*t2^2+2*p2*t1*t3^2-2*p1*r1*r3^2+2*p2*r1*r2^2

    d = 2*p1*t1*t2^2+2*p2*t1*t3^2+p1*r1*r3^2-4*p2*r1*r2^2 ......................(14)

    Define L4(p1,p2,r1,r2,r3,t1,t2,t3) with Left hand side of (13).

    It is understood that L4(1, 7, 37, 13, 36, 19, 13, 25) = 0 by the values of A0, B0, C0 and D0.
    
    L4(p1, 7, 37, 13, 36, 19, 13, 25) = 2032688644*p1^2 -22781495408*p1 + 20748806764 = 0

    Above solution are 1 and 741028813/72596023.

    So, L4(741028813/72596023, 7, 37, 13, 36, 19, 13, 25) = 0

    We get a new solution by using (14).
   
    a = 1058103081810 
    b = 535945811334 
    c = -1140105961325
    d = 944080652640 
    a+b+c+d =1398023584459
   




    L4(1, 7, r1, 13, 36, 19, 13, 25) = 4211236*r1^2  - 875076464*r1 + 26612647084 = 0

    Above solution are 37 and 9463957/55411.

    So, L4(1, 7, 9463957/55411, 13, 36, 19, 13, 25) = 0


    In the same way,we get a new solution by using (14).
   
    a = 378573600
    b = 145514934
    c = 65167315
    d = -201317790
    a+b+c+d = 387938059
    

    This solution is smaller than Jacobi and Madden's one.


4. Table of Solutions

Simcha Brudno, A further example of A^4 + B^4 + C^4 + D^4 = E^4, Proc. Camb. Phil. Soc. 60 (1964) 1027-1028.

Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4, Amer. Math. Monthly 115 (2008) 220-236.

Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000