1.Introduction

I show that there are infinitely many solutions for x1^k+x2^k+x3^k+x4^k+x5^k+x6^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k,
for k=2,4,6,8 by Sinha's Theorem.


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

      x1^k+x2^k+x3^k+x4^k+x5^k+x6^k+x7^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k+y7^k,for k=1,2,4,6,8.

      However,if x1+y1=0, we can obtain the solution for k.6.6.

2. Theorem


       There are infinitely many solutions for x1^k+x2^k+x3^k+x4^k+x5^k+x6^k =
       y1^k+y2^k+y3^k+y4^k+y5^k+y6^k,for k=2,4,6,8.


      x1 = bx+19a-5b
      x2 = (-a+b)x-14a+2b
      x3 = (b-3a)x+4a+4b
      x4 = -ax-9a-b
      x5 = (-2a+b)x+5a-3b
      x6 = 2ax-10a+6b
      
      y1 = (-2a+b)x+b+9a
      y2 = (b-3a)x-4a-4b
      y3 = (-a+b)x-2b+14a
      y4 = ax-19a+5b
      y5 = bx+3b-5a
      y6 = 2ax-6b+10a

      where (8x^2+21x-275)a^2+(-5x^2-24x+170)ab+(3x-3)b^2=0.


        

              
Proof.

     x1^k+x2^k+x3^k+x4^k+x5^k+x6^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k,for k=2,4,6,8...........(1)


     1. Solving for a1^2 +a2^2 +a3^2 = b1^2 +b2^2 +b3^2.


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


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


           a1^2 +a2^2 +a3^2 - ( b1^2 +b2^2 +b3^2) = 0



      2. Solving for a1^4 +a2^4 +a3^4 = b1^4 +b2^4 +b3^4.

      
           a1^4 +a2^4 +a3^4 -( b1^4 +b2^4 +b3^4)=-32x(a+b)(3a-b)f,

       
           f=(8x^2+21x-275)a^2+(-5x^2-24x+170)ab+(3x-3)b^2=0..............................(2)

      

     We must find rational value (a,b,x) for above equation.
     y^2=25x^4+144x^3-1280x^2-4608x+25600.................................................(3)

     Transform (3) to minimal Weierstrass form (4).

     V^2 = U^3 + U^2 -920U + 10404........................................................(4)

     We find that Rank of (4) is 2 and generator is {(16,-6) , (214,-3108)} by using MWRANK(Cremona).

     Since (4) has infinitely many rational points (U,V), (3) has infinitely many rational points (x,y).

     So,substitute (x,a,b) to (5), then we obtain infinitely many solutions of (1).

      
      x1 = 2a2 = bx+19a-5b
      x2 = b1+b2+b3 = -ax-14a+2b+bx
      x3 = 2a3 = bx+4a+4b-3ax
      x4 = b1-b2+b3 = -ax-9a-b
      x5 = -b1+b2+b3 = -2ax+5a-3b+bx
      x6 = b1+b2-b3 = 2ax-10a+6b
      

      y1 = -a1+a2+a3 = -2ax+b+9a+bx
      y2 = 2b3 = bx-3ax-4a-4b
      y3 = a1+a2+a3 = -ax-2b+14a+bx
      y4 = 2b1 = ax-19a+5b
      y5 = 2b2 = bx+3b-5a
      y6 = a1+a2-a3 = 2ax-6b+10a...........................................................(5)


     

    Q.E.D.



3. Example

        (8x^2+21x-275)a^2+(-5x^2-24x+170)ab+(3x-3)b^2=0

        (a,b)<500,height(x)<50

        [x, a, b  ]       [ x1    x2    x3     x4     x5    x6]  [ y1     y2    y3    y4     y5     y6]

        [1,     47,  82], [ 565, -459,  457,  -552,  -23,  116], [ 493,  -575,  529, -436,   93,    72]
        [6,     15, 139], [ 106,  203,  295,   -91,   78,  216], [ 232,   -13,  169,  125,  294,  -126]
        [16/3, 117,  83], [ 211, -155,  -59,  -165,  -44,   54], [  31,  -209,  121, -111,   10,   180]
        [29/6,   9,   2], [1024, -935, -461,  -759, -230,   54], [  34,  -989,  529, -705, -176,   990]
        [29/6,  46, 241], [5003, 4683, 9875, -5264, 1363, 8584], [8251, -3901, 6627, 3320, 9947, -3248]
        [16/7,   1, -25], [  19,  -27,  -35,     3,    4,  -34], [ -17,     7,    1,  -31,  -30,    36]
        [16/7,  63, 121], [ 190, -111,  127,  -182,  -13,   84], [ 148,  -195,  169,  -98,   71,    42]
        [38/25,  5,   9], [ 398, -287,  293,  -385,  -22,  120], [ 328,  -407,  363, -265,   98,    70]