1.Introduction

There are many solutions for A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46.

I show that there are infinitely many solutions for 6.4.4 by different from Crussol's method,
and describe a method of generating solutions.

[1]:Dickson: History of the Theory of Numbers V2


2. Theorem


       There are infinitely many solutions for A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46.

             A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46.................(1)


             A1=(3q+p)d+(q-p)c
             A2=(q+p)d+(q-3p)c
             A3=(p-q)d+(3q+p)c
             A4=(3p-q)d+(q+p)c
             B1=(3q-p)d+(q+3p)c
             B2=(q-p)d+(q+p)c
             B3=(q+p)d+(p-q)c
             B4=(q+3p)d+(p-3q)c



Proof.

      We use above Crussol's parameter.([1]:Dickson)

      A16+ A26+ A36+ A46 -( B16+ B26+ B36+ B46)
      =480qp(p2+q2)(-2c+d)(2d+c)(c2+d2)(3d2q2-3d2p2-4dp2c+4dq2c+6dqcp-3c2q2+3c2p2)

     (3d2q2-3d2p2-4dp2c+4dq2c+6dqcp-3c2q2+3c2p2)=(3q2-3p2)d2+(-4p2+4q2+6qp)cd+(-3q2+3p2)c2 = 0

     We must find rational value (c,d) for above equation.
     Discriminant=13q4-17q2p2+12q3p+13p4-12p3q=y2...................(2)
     So, we must find rational numbers p,q,y.

     U=q/p
     V=y/p2 

     V2 = 13U4+12U3-17U2-12U+13.....................................(3)

     
     Using APECS program by Ian Connell,Weierstrass form is

     Y2 = X3+X2-916X+9920...........................................(4)

     U = (X-128-Y)/(-Y+7X-32)
     V = (-864Y+144X2+2844X-103488+3X3)/(-Y+7X-32)2.................(5)

     X = (6V+8-22U+32U2)/(U2-2U+1)
     Y = (-6V+120-210U+6U2+42VU+192U3)/(U3-3U2+3U-1)................(6)

     Point P(1/13,45/13) satisfys (3).
     Rational point Q(X,Y) on the curve (4) corresponding to the values U=1/13,V=45/13 is
     X=32,Y=-120 by using (6).

     So, we get the relation of the curve (3) and the curve (4).

     Point P(1/13,45/13) on the curve (3) <=======>  Point Q(32,-120) on the curve (4)

     We obtain 2Q=(329/16, 909/64) on the curve (4) using APECS.
    

     As this point on the curve (4) does not have intger coordinates,
     there are infinitely many rational points on the curve (4) by Nagell-Lutz theorem.


 
     Point 2P=(-173/139,60051/19321) is given by 2Q using (5).

     We can get a new solution by point 2P=(-173/139,60051/19321).  

     138066+ 394576+ 319246+ 138176 = 294236+ 37726+218796+ 399866

     We can obtain infinitely many integer solutions  for (1) by apllying the group law.



    Q.E.D.