1.Introduction

We show a simple parametric solution of m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3).
See m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3)

This parametric solution gives infinite integral solutions for any m,n.

2.Theorem
     
    There is a parametric solution of m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3).
    
    x1 = nr+s+q
    x2 = nr-s-q
    x3 = -nr+s-q
    x4 = -nr-s+q
    y1 = ms+r+q
    y2 = -ms+r-q
    y3 = -ms-r+q
    y4 = ms-r-q
       
    m,n,q,r,s are arbitrary.
 
Proof.

m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3)..................(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 = ndef..........................................(3)

Let c=q and f=q.

Obviously we know the solution of equation (3) below.
a=nr, b=s, d=ms, e=r...........................................(4) 

Substitute above results (4) to (2) and simplifying (2), we obtain a parametric solution.

Q.E.D.　
 
　　                  
       
3.Example

(m,n)=(3,2)

(x1,x2,x3,x4) = ( 2r+s+q, 2r-s-q, -2r+s-q, -2r-s+q )
(y1,y2,y3,y4) = ( 3s+r+q, -3s+r-q, -3s-r+q, 3s-r-q )
q,r,s are arbitrary.

In this way, this parametric solution gives infinite solutions.