1.Introduction

There are many solutions for A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.

I show that there are infinitely many solutions for 6.6.6,and describe a method of generating solutions.




2. Theorem


       There are infinitely many solutions for A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.

             A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66..................(1)

        

             A1=qd-(p-q)c
             A2=(p+q)d-pc 
             A3=(p-q)d+qc
             A4=pd+(p+q)c
             A5=(p+2q)d-(2p-q)c
             A6=(2p-q)d+(p+2q)c
             B1=qd+pc
             B2=(p-q)d-(p+q)c
             B3=(p+q)d+(p-q)c
             B4=pd-qc
             B5=(2p-q)d-(p+2q)c
             B6=(p+2q)d+(2p-q)c
Proof.

      A16+ A26+ A36+ A46+ A56+ A66 -( B16+ B26+ B36+ B46+ B56+ B66)
      =15(c2+d2)dc(-2p+q)(p+2q)*(p2+q2)(-12q2c2+12q2d2+34pd2q-3qdpc-34pc^2q+12p2c2-12p2d2)


      (-12q2c2+12q2d2+34pd2q-3qdpc-34pc2q+12p2c2-12p2d2)
     =(34pq+12q2-12p2)d2-3qdpc+(-12q2+12p2-34pq)c2
     = 0

     We must find rational value (c,d) for above equation.
     Discriminant=3481p2q2+3264pq3-3264p3q+576q4+576p4=y2........................(2)
     So, we must find rational numbers p,q,y.

     U=q/p
     V=y/p2 

     V2 = 576U4+3264U3+3481U2-3264U+576..........................................(3)

     
     Using APECS program by Ian Connell,Weierstrass form is

     Y2+XY = X3-1001245X+ 385535537..............................................(4)

     U = (48X-27636)/(2Y+69X-39304)
     V = (14688Y-165816X2+96099228X-18682009464+96X3)/((2Y+69X-39304)2)..........(5)

     X = (12V+288-816U+290U2)/U2
     Y = (288V+6912-29520U+21294U2-414VU+9647U3)/U3..............................(6)

     Point P(0,-24) satisfys (3).
     Rational point Q(X,Y) on the curve (4) corresponding to the values U=0,V=-24 is
     X=2303/4,Y=-2915/8 by using (6).

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

     Point P(0,-24) on the curve (3) <=======>  Point Q(2303/4,-2915/8) on the curve (4)

     We obtain 2Q=(166193/256, -13782079/4096) 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=(-128/45,16712/675) is given by 2Q using (5).

     We can get the new solution by point 2P=(-128/45,16712/675).  

     44336+ 6866+ 131856+ 60646+ 37476+ 192496 = 71216+ 20026+ 122406+ 51196+ 31176+ 193616

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



    Q.E.D.