1.Introduction

A. Bremner and M. Ulas([1]) gave a parametric solution of  x^6 + y^3 = z^6 + w^3.

We show a parametric solution for px^6 + qy^3 = qz^6 + pw^3.


2.Theorem
      
 
There is a parametric solution of px^6 + qy^3 = qz^6 + pw^3,

      x = a(qb^6+pa^6)

      y = (qb^6+pa^6)b^2(-2pa^6+qb^6)

      z = b(qb^6+pa^6)

      w = -(qb^6+pa^6)a^2(2qb^6-pa^6).
    
     a, b, p, q: arbitrary



     
Proof.

px^6 + qy^3 = qz^6 + pw^3......................................(1)

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

(q-pm^3)t^3+(-3pa^2m^2+3qb^2)t^2+(-3pa^4m+3qb^4)t=0

Then we obtain m = qb^4/(pa^4) and t = -3b^2pa^6/(qb^6+pa^6).

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

 
Q.E.D.
　


3.Example



Case. (p,q,b)=(1,1,1)

x = a(1+a^6)
y = (1+a^6)(-2a^6+1)
z = 1+a^6
w = -(1+a^6)a^2(2-a^6)

This solution is same as Bremner and Ulas's one.



4.Reference

[1]. A. Bremner, M. Ulas, On x^a± y^b± z^c± w^d= 0, 1/a + 1/b + 1/c + 1/d = 1 , Int. J. NumberTheory, 7 (2011), no. 8