1.Introduction

F. Izadi and A. Shamsi Zargar([1]) showed that x^5 + y^3 = z^5 + w^3 has infinitely many integer solutions.

We show a simple parametric solution of x^5 + y^3 = z^5 + w^3 as a special case of px^n + qy^3 = qz^m + pw^3.

About the details of px^n + qy^3 = qz^m + pw^3, please see px^n + qy^3 = qz^m + pw^3. .


2.Theorem
      
 
There is a parametric solution of x^5 + y^3 = z^5 + w^3,

      x = a^3*(a^15+b^15)^3

      y = -(a^15+b^15)^4*b^5*(2*a^15-b^15)

      z = b^3*(a^15+b^15)^3

      w = (a^15+b^15)^4*a^5*(-2*b^15+a^15).
    
     a, b: arbitrary


     
Proof.

px^n + qy^3 = qz^m + pw^3.....................................(1)

Set (m,n,p,q)=(5,5,1,1) and M=15.

Set x = a^3, y = t+b^5, z = b^3, w = rt+a^5...................(2)

(1-r^3)t^3+(-3a^5r^2+3b^5)t^2+(-3a^10r+3b^10)t=0

Then we obtain r = b^10/(a^10) and t = -3b^5a^15/(a^15+b^15).

Substitute r and t to (2), and we obtain a parametric solution.

 
Q.E.D.
　



3.Reference

[1]. F. Izadi and A. Shamsi Zargar, On integer solutions of A^5 + B^3 = C^5 + D^3 , Notes on Number Theory and Discrete Mathematics, 7 (2014), vol. 20.