1.Introduction

I show that x1^k + x2^k + x3^k + x4^k + x5^k - y1^k - y2^k - y3^k - y4^k = 1   
for k=1,3,5,7 has infinitely many integer solutions.


We can get the solutions by using Sinha's theorem,

Sinha's theorem,
      a1n + a2n + a3n = b1n + b2n + b3n, (n=2,4)  and
      a1 + a2 - a3 =2(b1 + b2 - b3), a1 + a2 <> a3, b1 + b2 <> b3
      then

      (2a1-3h)n + (2a1-h)n + (2a2-3h)n + (2a2-h)n + (2b3+h)n
    = (2a3+h)n + (2a3+3h)n + (2b1-h)n + (2b2-h)n + (3h)n, 
      (n=1,3,5,7) where 2h=b1 + b2 - b3..........................................(0)

I show two solutions,but I guess that there are more solutions.



[1].Wolfram: Multigrade Eauation


2. Solutions

         
  
         1. (5x+3y)^n + (3x+y)^n + (-2x+1)^n + (2y+1)^n + (11x-y)^n - (2x+2y+1)^n - (7x+2y)^n - (5x-2y)^n - (3x+3y)^n = 1,
            for n=1,3,5,7 has infinitely many integer solutions where -6y^2-2y-1-30xy+81x^2 = 0.

               Denote s(n)=a1^n + a2^n + a3^n-(b1^n + b2^n + b3^n).

               a1 = x+3y+1
               a2 = x+y-1
               a3 = -6x-4y
               b1 = 8x+3y
               b2 = 6x-y
               b3 = 10x-2y
  
               s(2) =-2(-6y^2-2y-1-30xy+81x^2).

               s(4) =-2(-6y^2-2y-1-30xy+81x^2)(20y^2+2y+1-6xy+87x^2).

               So,if -6y^2-2y-1-30xy+81x^2 = 0 then this is a solution of (0) by Sinha's theorem.

               -6y^2-2y-1-30xy+81x^2 = 0........................................(1)
               Discriminant of (1) = 2844 > 0 and is not a perfect square,then (1) has infinitely many
               integer solutions.(Gauss's theorem?)
               Some small solutions of (1) are (x,y)=(-49,340), (-25009,-48620),
               (-1253761,8706238), (-640179841,-1244571842), and (-32093773489,222862282900). 

               We obtain

               x1...x5=[-2x+1, 2y+1, -2x-2y-1, -1, 11x-y] 

               y1...y5=[-5x-3y, -3x-y, 7x+2y, 5x-2y, 3x+3y] 
               
               Case of (x,y)=(-49,340)

               99^n + 681^n + 775^n + 193^n + 925^n - 583^n - 879^n - 337^n - 873^n = 1
                                                                            (n=1,3,5,7)

               Case of (x,y)=(-25009,-48620)

               50019^n + 147257^n + 272303^n + 27805^n + 220887^n
             - 97239^n - 226479^n - 270905^n - 123647^n = 1

               
               Case of (x,y)=(-1253761,8706238)
           
               2507523^n + 17412477^n + 19849909^n + 4944955^n + 23681281^n
             - 14904955^n - 22497609^n - 8636149^n - 22357431^n = 1

               Case of (x,y)=(-640179841,-1244571842)

               1280359683^n + 6970402571^n + 711755521^n + 5654255049^n + 3769503365^n 
             - 2489143683^n - 5797406409^n - 6934614731^n - 3165111365^n = 1


               Case of (x,y)=(-32093773489,222862282900)

               64187546979^n + 445724565801^n + 508117981255^n + 126580962433^n + 606193433245^n
             - 381537018823^n - 575893791279^n - 221068151377^n - 572305528233^n = 1

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             We can construct integer solutions infinitely.
          
         2. (8x+2y+1)^n + (2x+1)^n + (11x+2y)^n + (x-2y)^n + (9x+3y)^n - (6x+2y+1)^n - (11x+3y)^n - (5x+y)^n - (9x-y)^n = 1
            for n=1,3,5,7 has infinitely many integer solutions where 15x^2+(58y+8)x+6y^2+2y+1 = 0.

               a1 = 9x+3y+1
               a2 = x+y-1
               a3 = -14x-4y
               b1 = 14x+3y
               b2 = 4x-y
               b3 = 6x-2y
            
               s(2) =2(15x^2+(58y+8)x+6y^2+2y+1).

               s(4) =2(15x^2+(58y+8)x+6y^2+2y+1)(167x^2+(78y+8)x+20y^2+2y+1).

               So,if 15x^2+(58y+8)x+6y^2+2y+1 = 0 then this is a solution of (0) by Sinha's theorem.
        
               15x^2+(58y+8)x+6y^2+2y+1  = 0........................................(2)
               Similarly, (2) has infinitely many integer solutions.
               Some small solutions of (2) are (x,y)=(7,-2), (7,-66), (-3,28), (4023,-1070), (-87683,23318),
               (-87683,824284), and (-3099549,824284).

             
                Case of (x,y)=(7,-2)

                73^n + 11^n + 57^n + 53^n + 15^n - 65^n - 71^n - 33^n - 39^n = 1
                                                                       (n=1,3,5,7)

                Case of (x,y)=(7,-66)

                89^n + 15^n + 121^n + 31^n + 139^n - 135^n - 75^n - 129^n - 55^n = 1


                Case of (x,y)=(-3,28)

                23^n + 33^n + 55^n + 57^n - 39^n - 5^n - 51^n - 59^n - 13^n = 1


                Case of (x,y)=(4023,-1070)

                42113^n + 30045^n + 8047^n + 6163^n + 32997^n
              - 37277^n - 41043^n - 19045^n - 21999^n = 1


                Case of (x,y)=(-87683,23318)

                479461^n + 812465^n + 894559^n + 415097^n
              - 654827^n - 175365^n - 917877^n - 134319^n - 719193^n = 1


                Case of (x,y)=(-87683,824284)

                1683705^n + 1613431^n + 947105^n + 684055^n
              - 175365^n - 1508339^n - 385869^n - 1736251^n - 1122471^n = 1


                Case of (x,y)=(-3099549,824284)

                31622187^n + 28720225^n + 16948725^n + 14673461^n
              - 23147823^n - 6199097^n - 32446471^n - 4748117^n - 25423089^n = 1

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