1.Introduction

I show that x1^k+x2^k+x3^k+x4^k+x5^k+x6^k+x7^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k+y7^k   
for k=1,2,4,6,8 has the parametric solutions by quadratic forms.

We can get the parametric 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].Wolfram: Multigrade Eauation


2. Solutions

         

         1. (x+3y+1)n + (x+y-1)n + (-6x-4y)n = (8x+3y)n + (6x-y)n + (10x-2y)n,(n=2,4)

            Denote sn=a1n+a2n+a3n-(b1n+b2n+b3n)

            s2 =-2(-6y2-2y-1-30xy+81x2)

            s4 =-2(-6y2-2y-1-30xy+81x2)(20y2+2y+1-6xy+87x2)

            So,if -6y2-2y-1-30xy+81x2 = 0 then this is a solution of 8.7.7 by Shinha's theorem.
   
   
           Parametric solution is 

           (-951+887k-162k^2)^n + (426+11k-100k^2)^n + (738-444k+108k^2)^n + (279-750k+226k^2)^n
           + (207+21k-16k^2)^n+ (732-877k+246k^2)^n + (-201+412k-122k^2)^n
           =
           (-549+63k+82k^2)^n + (828-813k+144k^2)^n + (939-856k+230k^2)^n + (-123+74k-18k^2)^n 
           + (6+433k-138k^2)^n+ (531-465k+124k^2)^n + (-402+824k-244k^2)^n
           (n=1,2,4,6,8) 



         2. (2x+y+1)n + (x+3y-1)n + (-9x-4y)n = (12x+3y)n + (9x-y)n + (15x-2y)n,(n=2,4)
            
            s2 = -2(-6y2+2y-1-44xy-x+182x2)

            s4 = -2(-6y2+2y-1-44xy-x+182x2)(20y2-2y+1-10xy+x+196x2)


            Parametric solution is  

            (-2558+967k+16k^2)^n + (-96+1095k-261k^2)^n + (1548-468k+162k^2)^n + (2138-1906k+191k^2)^n
            + (46+301k+13k^2)^n+ (2700-1761k+258k^2)^n + (-1198+992k-109k^2)^n 
            =
            (-162-1017k+234k^2)^n + (2300-889k-43k^2)^n + (2746-1460k+271k^2)^n + (-258+78k-27k^2)^n
            + (-1152+1293k-96k^2)^n+ (1502-769k+149k^2)^n + (-2396+1984k-218k^2)^n
            (n=1,2,4,6,8) 

         3. (4x+2y+1)^n + (x+2y-1)^n + (-7x-4y)^n = (7x+3y)^n + (2x-y)^n + (3x-2y)^n,(n=2,4)

            s2 = 4x^2+50xy+6x+10y^2+2

            s4 = 2(2x^2+25xy+3x+5y^2+1)(40x^2+3x+1+35xy+19y^2)


            Parametric solution is

            (-2+35k^2+27k)^n + (-3+40k^2-23k)^n + (-6-30k^2)^n + (7-65k^2-4k)^n
            + (-4+5k^2+k)^n+ (1-70k^2-3k)^n + (-3+35k^2+2k)^n
            =
            (4-35k^2+23k)^n + (3-30k^2-27k)^n + (-3-65k^2-2k)^n + (1+5k^2)^n
            + (-7+40k^2+3k)^n+ (-2-35k^2-k)^n + (-6+70k^2+4k)^n
            (n=1,2,4,6,8) 

         4. (7x+y+1)^n + (3x+3y-1)^n + (-14x-4y)^n = (14x+3y)^n + (4x-y)^n + (6x-2y)^n,(n=2,4)

            s2 = 2(3x^2+4x+1+46xy-2y+6y^2)
            s4 = 2(3x^2+4x+1+46xy-2y+6y^2)(155x^2+4x+1+66xy-2y+20y^2)
            
           

            Parametric solution is

            (-2+31k+6k^2)^n + (-3-17k+60k^2)^n + (-6+12k-36k^2)^n + (7-18k-54k^2)^n
            + (-4+9k)^n+ (1-5k-66k^2)^n + (-3+8k+30k^2)^n 
            =
            (4+15k-54k^2)^n + (3-33k)^n + (-3+4k-66k^2)^n + (1-2k+6k^2)^n
            + (-7+17k+30k^2)^n+ (-2+3k-36k^2)^n + (-6+16k+60k^2)^n
            (n=1,2,4,6,8)









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