1.Introduction

I proved that there was always a parametric solution of x1^4 + x2^4 +...+ xn^4 = nw^2.



2.Theorem
      
     
        
    n:integer

  If n > 2, there is always a parametric solution of x1^4 + x2^4 +...+ xn^4 = nw^2.




     
Proof.
a1,a2,...,an:integer

(x+a1)^4 + (x+a2)^4 +...+ (x+an-1)^4 + (x-a1-a2-...-an-1)^4 - n(x^2+h)^2............................(1)


Expanding and simplifying (1),we obtain
f = f2x2 + f1x + f0
fi:coefficient of xi

First,solve f2=0 for h.

Next,solve f1x + f0 =0 for x,and obtain
     x=-f0/f1
            
Substitute h,x to (1),and obtain a parametric solution.

   
Q.E.D. 
 
                    
       
3.Example

I show two examples.



    
Case n=3

     x1= 9a1^2a2^2+10a1a2^3+4a1^3a2+5a2^4+5a1^4

     x2= 9a1^2a2^2+4a1a2^3+10a1^3a2+5a2^4+5a1^4

     x3= 27a1^2a2^2+16a1a2^3+16a1^3a2+5a2^4+5a1^4 

      w= (a2^2+a1a2+a1^2)*(25a2^6+75a2^5a1+222a2^4a1^2+319a2^3a1^3+222a2^2a1^4+75a2a1^5+25a1^6)


Case n=4

     x1= 7a1^4+7a2^4+7a3^4+12a1a2^2a3+12a1a2a3^2+2a1^3a2+2a1^3a3+9a1^2a2^2+9a1^2a3^2+14a1a2^3
       + 14a1a3^3+14a2a3^3+14a2^3a3+21a2^2a3^2

     x2= 7a1^4+7a2^4+7a3^4+12a1a2a3^2+14a1^3a2+14a1^3a3+9a1^2a2^2+21a1^2a3^2+2a1a2^3+14a1a3^3
       + 14a2a3^3+2a2^3a3+9a2^2a3^2+12a1^2a2a3

     x3= 7a1^4+7a2^4+7a3^4+12a1a2^2a3+14a1^3a2+14a1^3a3+21a1^2a2^2+9a1^2a3^2+14a1a2^3+2a1a3^3
       + 2a2a3^3+14a2^3a3+9a2^2a3^2+12a1^2a2a3

     x4= 7a1^4+7a2^4+7a3^4+72a1a2^2a3+72a1a2a3^2+26a1^3a2+26a1^3a3+45a1^2a2^2+45a1^2a3^2
       + 26a1a2^3+26a1a3^3+26a2a3^3+26a2^3a3+45a2^2a3^2+72a1^2a2a3


     w= 706a1^2a3^6+1432a1^5a2^3+1432a1^3a2^5+196a1^7a2+706a1^6a2^2+1432a3^5a2^3+706a3^6a2^2
      + 196a3^7a2+1432a3^3a2^5+1795a3^4a2^4+1795a1^4a2^4+706a3^2a2^6+706a1^2a2^6+196a1a3^7
      + 196a1a2^7+196a3a2^7+1432a1^5a3^3+3540a1^5a3a2^2+3540a1^5a3^2a2+1160a1^6a3a2
      + 11504a1^3a3^3a2^2+11504a1^3a3^2a2^3+5940a1a3^3a2^4+3540a1^2a3^5a2+5940a1^4a3a2^3
      + 8712a1^4a3^2a2^2+3540a1a3^2a2^5+11504a1^2a3^3a2^3+1160a1a3a2^6+8712a1^2a3^2a2^4
      + 1160a3^6a1a2+8712a1^2a3^4a2^2+5940a1a3^4a2^3+5940a1^4a3^3a2+3540a1^2a3a2^5+49a2^8
      + 49a3^8+1795a1^4a3^4+196a1^7a3+1432a3^5a1^3+706a1^6a3^2+49a1^8+3540a3^5a1a2^2
      + 5940a3^4a1^3a2+5940a1^3a3a2^4


 














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