1.Introduction

It's showed that x^4 + y^4 = z^3 + w^3 has infinitely many integral solutions on MathOverflow  website[1].

We show the parametric solutions for x^n + y^n = z^(n-1) + w^(n-1).


2.Theorem
 

There is a parametric solution of x^n + y^n = z^(n-1) + w^(n-1),

      x = p(r^(n-1)+s^(n-1))(p^n+q^n)^(M/n-1)

      y = q(r^(n-1)+s^(n-1))(p^n+q^n)^(M/n-1)

      z = r(r^(n-1)+s^(n-1))(p^n+q^n)^(M/(n-1)-1)

      w = s(r^(n-1)+s^(n-1))(p^n+q^n)^(M/(n-1)-1).

 condition: 
    
     M=lcm(n,n-1)
     n,p,q,r,s are arbitrary.

     
Proof.

x^n + y^n = z^(n-1) + w^(n-1).....................(1)
Let M = lcm(n,n-1).
Set x=pt, y=qt, z=rt, w=st........................(2)

Then we obtain t = (r^(n-1)+s^(n-1))/(p^n+q^n).
Substitute t to equation (2), and we obtain a solution.


 
Q.E.D.


3.Examples


Case. n=4

x = p(r^3+s^3)(p^4+q^4)^2
y = q(r^3+s^3)(p^4+q^4)^2
z = r(r^3+s^3)(p^4+q^4)^3
w = s(r^3+s^3)(p^4+q^4)^3

Case. n=5

x = p(r^4+s^4)(p^5+q^5)^3
y = q(r^4+s^4)(p^5+q^5)^3
z = r(r^4+s^4)(p^5+q^5)^4
w = s(r^4+s^4)(p^5+q^5)^4




4.Reference

[1]: MathOverflow: a^2+b^2=c^3+d^3=......=x^k+y^k