1.Introduction

I show that ( 3+4a )^n + ( a+2a^2-2 )^n + ( 1+5a+2a^2 )^n - ( 3a+2a^2+2 )^n - ( 3+3a+2a^2 )^n = 1 
for k=2,4 has infinitely many integer solutions.




2. Solutions

         x1^n + x2^n + x3^n = y1^n + y2^n + y3^n for k=2,4 has a paramertic solution as follows.

         x1 = a1x-a1-2a2             y1 = a1x+a1+2a2
         x2 = a2x+2a1+a2             y2 = a2x-2a1-a2
         x3 = (a1+a2)x+a1-a2         y3 = (a1+a2)x-a1+a2.............................(1)
 
         You can see detail information at this page.
         A^4+ B^4+ C^4= D^4+ E^4+ F^4 PART 2

         Let x1=1,then x=(a1+2a2+1)/a1.
         Now,x must be integer,so set a1=1 in a simple way,then we get x=2(a2+1).....(2)

         Substitute (2) to (1) and replace variable a2 to a,then we get

         ( 3+4a )^n + ( a+2a^2-2 )^n + ( 1+5a+2a^2 )^n - ( 3a+2a^2+2 )^n - ( 3+3a+2a^2 )^n = 1
         for k=2,4 has infinitely many integer solutions.............................(3)
            
         For example,

             a    (n=2,4)
              
             2    11^n +   8^n +  19^n -  16^n -  17^n = 1 
             3    15^n +  19^n +  34^n -  29^n -  30^n = 1
             4    19^n +  34^n +  53^n -  46^n -  47^n = 1
             5    23^n +  53^n +  76^n -  67^n -  68^n = 1
             6    27^n +  76^n + 103^n -  92^n -  93^n = 1
             7    31^n + 103^n + 134^n - 121^n - 122^n = 1
             8    35^n + 134^n + 169^n - 154^n - 155^n = 1
             9    39^n + 169^n + 208^n - 191^n - 192^n = 1
            10    43^n + 208^n + 251^n - 232^n - 233^n = 1
            11    47^n + 251^n + 298^n - 277^n - 278^n = 1
            12    51^n + 298^n + 349^n - 326^n - 327^n = 1
            13    55^n + 349^n + 404^n - 379^n - 380^n = 1
            14    59^n + 404^n + 463^n - 436^n - 437^n = 1
            15    63^n + 463^n + 526^n - 497^n - 498^n = 1
            16    67^n + 526^n + 593^n - 562^n - 563^n = 1
            17    71^n + 593^n + 664^n - 631^n - 632^n = 1
            18    75^n + 664^n + 739^n - 704^n - 705^n = 1
            19    79^n + 739^n + 818^n - 781^n - 782^n = 1
            20    83^n + 818^n + 901^n - 862^n - 863^n = 1