1.Introduction

I show that A1^k + A2^k + A3^k - B1^k - B2^k - B3^k - B4^k = 1   
for k=2,4,6 has infinitely many integer solutions.


We can get the parametric solutions by using Chernick's theorem,

Chernick's theorem,

     (a-7)^n + (a-2b+1)^n + (3a+1)^n + (3a+2b+1)^n = (a+7)^n + (a-2b-1)^n + (3a-1)^n + (3a+2b-1)^n
     for n= 2,4,6 where a^2+ab+b^2=7.





2. Solutions

         Chernick's identity has a paramertic solution as follows.

         x1= 5x^2+9xy+10y^2              y1= 9x^2+5xy+4y^2
 
         x2= 9x^2+7xy-4y^2               y2= 7x^2+5xy-6y^2

         x3= 7x^2-5xy-8y^2               y3= 5x^2-7xy-10y^2
      
         x4= x^2-13xy-6y^2               y4= x^2+15xy+8y^2


  
         1. y1^n + y2^n + y3^n + y4^n - x1^n - x3^n - x4^n = 1,
            for n=2,4,6 has infinitely many integer solutions where 9x^2+7xy-4y^2=1.

              
               
           9x^2+7xy-4y^2=1........................................(1)
           Discriminant of (1) = 193 > 0 and is not a perfect square,then (1) has infinitely many integer solutions.
           Some small solutions of (1) are (x,y)=(-65, 56), and (3969297065, 10366039576). 

               
               
           Case of (x,y)=(-65,56)

           32369^n + 7441^n + 15245^n + 25287^n - 19725^n - 22687^n - 32729^n = 1
                                                                            (n=2,4,6)
                          

           Case of (x,y)=(3969297065, 10366039576)

           777346431000204789329^n + 328711972292213036881^n + 1283792402213729037715^n + 1492581888090163359033^n
          -1523637375045286468845^n - 955080429921516000833^n - 1163869915797950322151^n = 1


                                                .
                                                .
                                                .
                                                .
                                                .
             We can construct integer solutions infinitely.
          
        2. y1^n + y2^n + y3^n + y4^n - x1^n - x2^n - x4^n = 1,
           for n=2,4,6 has infinitely many integer solutions where 7x^2-5xy-8y^2=1.

              

           7x^2-5xy-8y^2=1........................................(2)
           Discriminant of (2) = 249 > 0 and is not a perfect square,then (2) has infinitely many integer solutions.
           Some small solutions of (2) are (x,y)=(3, 2), (-183873, 238802),and (51139017, 34454062). 

                
           Case of (x,y)=(3, 2)

           127^n + 69^n + 37^n + 131^n - 139^n - 107^n - 93^n = 1
                                                            (n=2,4,6)


           Case of (x,y)=(-183873, 238802)

           312842901247^n + 325039611051^n + 93852870373^n + 168618160429^n
          -344127191371^n - 231186740677^n - 262471030803^n = 1 


           Case of (x,y)=(51139017, 34454062)

           37094855402421247^n + 19993633399970229^n + 11128456620726373^n + 38541061101180851^n
          -40804340942663371^n - 31122090020696603^n - 27412604480454477^n = 1


           Case of (x,y)=(3145631250987, -4085336259262)

           91560046070013292292448127^n + 95129669376767097692346651^n + 27468013821003987687734437^n
          +49349646341760451546122589^n - 100716050677014621521692939^n - 67661655555763110004612213^n
          -76817660162764439233857027^n = 1