1.Introduction

There are many solutions for A18+ A28+ A38+ A48+ A58+ A68 = B18+ B28+ B38+ B48+ B58+ B68.

I show that there are infinitely many solutions for 8.6.6 by Sinha's Theorem.

      By Sinha's Theorem( See the previous problem [61])

      A18+ A28+ A38+ A48+ A58+ A68+ A78 = B18+ B28+ B38+ B48+ B58+ B68+ B78.

      But,if A1+B1=0,we can obtain the solution for 8.6.6.

2. Theorem


       There are infinitely many solutions for A18+ A28+ A38+ A48+ A58+ A68 = B18+ B28+ B38+ B48+ B58+ B68.

             A18+ A28+ A38+ A48+ A58+ A68 = B18+ B28+ B38+ B48+ B58+ B68...................(0)

        

             
Proof.

     1. Solving for a12 +a22 +a32 = b12 +b22 +b32.

         Set a1=ax+s1,a2=bx+s2,a3=bx+s2-3(ax+s1),b1=ax-s2,b2=bx-s3,b3=(b-3a)x-s2+3s1...(1)

         Take s1=5a-3b,s2=19a-5b then

         a12 +a22 +a32 - ( b12 +b22 +b32) = 0


     2. Solving for a14 +a24 +a34 = b14 +b24 +b34.
      
        a14 +a24 +a34 -( b14 +b24 +b34)=-32x(a+b)(3a-b)f
       
        f=(8x2+21x-275)a2+(-5x2-24x+170)ab+(3x-3)b2
      

     We must find rational value (a,b) for above equation.
     Discriminant=25x4+144x3-1280x2-4608x+25600=y2.............................(2)
     So, we must find rational numbers x,y.

     U=x
     V=y

     V2 = 25U4+144U3-1280U2-4608U+25600........................................(3)

     
     Using APECS program by Ian Connell,Weierstrass form is

     Y2=X3+X2-920X+10404.......................................................(4)

     U = (200X-3248)/(5Y+9X-162)
     V = (25344Y-194880X2+3550080X-23468800+4000X3)/(5Y+9X-162)2...............(5)

     X = (5V+800-72U-7U2)/U2
     Y = (200V+32000-4320U-800U2-9VU+45U3)/U3..................................(6)

     Point P(0,160) satisfys (3).
     Rational point Q(X,Y) on the curve (4) corresponding to the values U=0,V=160 is
     X=406/25,Y=-396/125.

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

     Point P(0,160) on the curve (3) <=======>  Point Q(406/25, -396/125) on the curve (4)

     We obtain 2Q=(4939/25, -344112/125) 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=(-200/67, 725280/4489) is given by 2Q using (5).

     We can obtain infinitely many integer solutions  for (2) by apllying the group law.

     From Sinha's Theorem

      A18+ A28+ A38+ A48+ A58+ A68+ A78 = B18+ B28+ B38+ B48+ B58+ B68+ B78.

      A1=2a1
      A2=2a2
      A3=b1+b2+b3
      A4=2a3
      A5=b1-b2+b3
      A6=-b1+b2+b3
      A7=b1+b2-b3

      B1=a1-a2+a3
      B2=-a1+a2+a3
      B3=2b3
      B4=a1+a2+a3
      B5=2b1
      B6=2b2
      B7=a1+a2-a3

     By Sinha's Theorem,substitute (a,b,x) to (1),then we obtain infinitely many solutions of (0).

    Q.E.D.



3. Example

( x, a, b ):(8x2+21x-275)a2+(-5x2-24x+170)ab+(3x-3)b2=0

( 1 47  82)  5658+ 4598+ 4578+ 5528+  238+ 1168= 4938+ 5758+ 5298+ 4368+  938+  728
(-6 21 113)  2118+ 1558+  598+  448+ 1658+  548=  318+ 2098+ 1218+  108+ 1118+ 1808
( 6 15 139)  1068+ 2038+ 2958+  918+  788+ 2168= 2328+  138+ 1698+ 1258+ 2948+ 1268
(-14 5   9)   198+  278+  358+   48+   38+  348=  178+   78+   18+  308+  318+  368
(-14 3 -37)  1908+ 1118+ 1278+  138+ 1828+  848= 1488+ 1958+ 1698+  718+  988+  428