1.Introduction

I show that A1^n + A2^n + ...+ A7^n - B1^n - B2^n - ... -B6^n = 1   
for n=1,2,4,6,8 has infinitely many integer solutions.


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

Sinha's theorem,

      if a1^n +a2^n +a3^n = b1^n +b2^n +b3^n (n=2,4) then
        
      (2a1)^n +(2a2)^n +(b1 +b2 +b3)^n +(2a3)^n +(b1 -b2 +b3)^n +(-b1 +b2 +b3)^n +(b1 +b2 -b3)^n

     =(a1 -a2 +a3)^n +(-a1 +a2 +a3)^n +(2b3)^n +(a1 +a2 +a3)^n +(2b1)^n +(2b2)^n +(a1 +a2 -a3)^n,
      (n=1,2,4,6,8)....................................................................(1)





[1].Wolfram: Multigrade Eauation


2. Solutions

         
  
           1.  y1^n + y2^n + y3^n + y4^n + y5^n + y6^n + y7^n
             - x1^n - x2^n - x3^n - x4^n - x5^n - x6^n = 1 for n=1,2,4,6,8,
               has infinitely many integer solutions where 2061y^2-110xy+x^2=1.
               x,y:integer

               y1= 955y^2-226xy+7x^2            x1= 7138y^2-556xy+10x^2
               y2= -6549y^2+510xy-9x^2          x2= -366y^2+180xy-6x^2
               y3= 5006y^2-340xy+6x^2           x3= 7067y^2-450xy+7x^2
               y4= 589y^2-46xy+x^2              x4= -5594y^2+284xy-2x^2
               y5= 1766y^2-36xy-2x^2            x5= -295y^2+74xy-3x^2
               y6= 7362y^2-524xy+10x^2          x6= 5301y^2-414xy+9x^2
               y7= 6183y^2-330xy+3x^2
               
               

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

               a1 = 5X+2Y+1
               a2 = X+2Y-1
               a3 = -6X-5Y
               b1 = 4X+5Y
               b2 = 2X-3Y
               b3 = 2X-Y
  
               s(2) =2(19X^2+(30Y+4)X-Y^2+1).

               s(4) =2(19X^2+(30Y+4)X-Y^2+1)(43X^2+(42Y+4)X+25Y^2+1).

               So,if 19X^2+(30Y+4)X-Y^2+1 = 0 then this is a solution of (1) by Sinha's theorem.
               We obtain,

               {x1,...,x7}= {7138y^2-556xy+10x^2, -366y^2+180xy-6x^2, 7067y^2-450xy+7x^2,
                            -5594y^2+284xy-2x^2, -295y^2+74xy-3x^2, 5301y^2-414xy+9x^2, 2061y^2-110xy+x^2}

               {y1,...,y7}= {955y^2-226xy+7x^2, -6549y^2+510xy-9x^2, 5006y^2-340xy+6x^2,
                             589y^2-46xy+x^2, 1766y^2-36xy-2x^2, 7362y^2-524xy+10x^2, 6183y^2-330xy+3x^2}

               2061y^2-110xy+x^2=1........................................(2)
               Discriminant of (2) = 964 > 0 and is not a perfect square,then (2) has infinitely many
               integer solutions.
               Some small solutions of (2) are (x,y)=(27950276695727749, 324820602522300), (7779989581725251, 324820602522300).
                           
               
               
               Case of (x,y)=(27950276695727749, 324820602522300)

                 3517471704289598081392597217788807^n
               + (-3091735258270818071238513145296009)^n
               + 2128682230093900050770420362464006^n
               + 425736446018780010154084072492801^n
               + (-1702945784075120040616336289971202)^n
               + 3831628014169020091386756652435210^n
               + 3^n
               - 3517471704289598081392597217788810^n
               - (-3091735258270818071238513145296006)^n
               - 2128682230093900050770420362464007^n
               - 425736446018780010154084072492798^n
               - (-1702945784075120040616336289971203)^n
               - 3831628014169020091386756652435209^n
               = 1,
               n=(1,2,4,6,8).



               Case of (x,y)=(7779989581725251, 324820602522300)

                (-46666594209834712899374365948793)^n
               + 53092652169857923724991193295991^n
               + 32130289800116054128084136736006^n
               + 6426057960023210825616827347201^n
               + (-25704231840092843302467309388802)^n
               + 57834521640208897430551446124810^n
               + 3^n
               - (-46666594209834712899374365948790)^n
               - 53092652169857923724991193295994^n
               - 32130289800116054128084136736007^n
               - 6426057960023210825616827347198^n
               - (-25704231840092843302467309388803)^n
               - 57834521640208897430551446124809^n
               = 1,
               n=(1,2,4,6,8).
               
              
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                                                .
             We can construct integer solutions infinitely.
        
             There is interesting relation between x and y such that abs(x-y) is 1 or 3.



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