1.Introduction

We show a parametric solution of m(x1^5+x2^5+x3^5+x4^5)=n(y1^5+y2^5+y3^5+y4^5).

See x1^n + x2^n + x3^n = y1^n + y2^n + y3^n, (n=1,5)

This parametric solution gives infinite integral solutions.

2.Theorem
     
    There is a parametric solution of m(x1^5+x2^5+x3^5+x4^5)=n(y1^5+y2^5+y3^5+y4^5).
    
    x1 = (v^2u^2-u^2-v^2+1)p+2vu^2+2v+2uv^2-2u
    x2 = (-v^2u^2+u^2+v^2-1)p+2vu^2+2v-2uv^2+2u
    x3 = (-v^2u^2+u^2+v^2-1)p-2vu^2-2v+2uv^2-2u
    x4 = (v^2u^2-u^2-v^2+1)p-2vu^2-2v-2uv^2+2u
    y1 = (v^2u^2-u^2-v^2+1)p+2uv^2+2u+2vu^2-2v
    y2 = (-v^2u^2+u^2+v^2-1)p-2uv^2-2u+2vu^2-2v
    y3 = (v^2u^2-u^2-v^2+1)p-2uv^2-2u-2vu^2+2v
    y4 = (-v^2u^2+u^2+v^2-1)p+2uv^2+2u-2vu^2+2v
       
    m = (v^2+1)(u^2-1), n = (u^2+1)(v^2-1)   
    p,u,v are arbitrary.


 
Proof.

m(x1^5+x2^5+x3^5+x4^5)=n(y1^5+y2^5+y3^5+y4^5)............................................(1)

Let (x1,x2,x3,x4)=(a+b+c, a-b-c, -a+b-c, -a-b+c)
    (y1,y2,y3,y4)=(d+e+f, -d+e-f, -d-e+f, d-e-f).........................................(2)

We obtain mabc(a^2+b^2+c^2) = ndef(d^2+e^2+f^2)..........................................(3)

Let c=p and f=p.

Substitute a=nr, b=s, d=ms, e=r to (3) and simplifying (3), we obtain

(n^2r^2+s^2-m^2s^2-r^2)=0................................................................(4)

Easily, we obtain the solution of equation (4) below.

s = 2u/(u^2-1), n = (u^2+1)/(u^2-1), r = 2v/(v^2-1), m = (v^2+1)/(v^2-1).................(5)

Substitute above results (5) to (2) and simplifying (2), we obtain a parametric solution.

Q.E.D.　
 
　　                  
       
3.Example

(u,v)=(3,2)
(m,x1,x2,x3,x4) = ( 4, 12p+29,-12p+11,-12p-11,12p-29 )
(n,y1,y2,y3,y4) = ( 3, 12p+31,-12p+1,12p-31,-12p-1 )

p is arbitrary.
In this way, this parametric solution gives infinite solutions.