1.Introduction

I show that A1^n + A2^n + A3^n + A4^n - B1^n - B2^n - B3^n = 1,   
for n=1,3,5 has infinitely many integer solutions.





[1].Wolfram: Multigrade Eauation


2. Solutions

         
  
           1.  x1^n + x2^n + x3^n + x4^n - y1^n - y2^n - y3^n = 1 for n=1,3,5, 
               has infinitely many integer solutions where 11xy+3y^2+x^2=1.

               x,y:integer

               x1 = -3xy-5y^2+5x^2            y1 = -5xy-7y^2+3x^2
               x2 = 13xy+5y^2+3x^2            y2 = 7xy-3y^2+5x^2
               x3 = 9xy-y^2+7x^2              y3 = 7xy+7y^2+7x^2
               x4 = xy+y^2+x^2            
               
               
                             

               11xy+3y^2+x^2 = 1........................................(1)
               Discriminant of (1) = 109 > 0 and is not a perfect square,then (1) has infinitely many
               integer solutions.
               Some small solutions of (1) are (x,y)=(-1826, 6525), (-69949, 6525), (-124392599, 444502575), and (-4765135726, 444502575).

                           
               
               
               Case of (x,y)=(-1826, 6525)

                  67990503^n + 33995251^n + 228453297^n + 194458045^n
                - 237966757^n - 160462795^n - 126467543^n = 1
                 (n=1,3,5).


               Case of (x,y)=(-69949, 6525)

                  25620686555^n + 8958042003^n + 30099707557^n + 4479021001^n
                - 16662644553^n - 21141665555^n - 31353147007^n = 1


               Case of (x,y)=(-124392599, 444502575)
             
                  315526454602326003^n + 157763227301163001^n + 1060193065381277847^n
                + 902429838080114845^n - 1104342591108141007^n - 744666610778951845^n
                - 586903383477788843^n = 1


               Case of (x,y)=(-4765135726, 444502575)

                  118899025041394065605^n + 41571971851903366503^n + 139685010967345748857^n
                + 20785985925951683251^n  - 77327053189490699103^n - 98113039115442382355^n
                - 145501901481661782757^n = 1


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             We can construct integer solutions infinitely.