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^n + qy^3 = qz^m + pw^3.


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

      x = a^(M/n)*(p*a^M+q*b^M)^(M/n)

      y = -3*(p*a^M+q*b^M)^(1/3*M-1)*b^(1/3*M)*p*a^M+(p*a^M+q*b^M)^(1/3*M)*b^(1/3*M)

      z = b^(M/m)*(p*a^M+q*b^M)^(M/m)

      w = -3*(p*a^M+q*b^M)^(1/3*M-1)*q*b^M*a^(1/3*M)+(p*a^M+q*b^M)^(1/3*M)*a^(1/3*M)
    
     a, b, p, q, m, n: arbitrary
     M=lcm(3,m,n)


     
Proof.

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

Set x = a^(M/n), y = t+b^(1/3M), z = b^(M/m), w = rt+a^(1/3M)...................(2)

(q-pr^3)t^3+(-3pa^(1/3M)r^2+3qb^(1/3M))t^2+(-3p(a^(1/3M))^2r+3q(b^(1/3M))^2)t=0

Then we obtain r = q(b^(2/3M))/(p(a^(2/3M))) and t = -3b^(1/3M)pa^M/(pa^M+qb^M).

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

 
Q.E.D.
　


3.Examples




Case. (m,n)=(6,6)

x = a*(p*a^6+q*b^6)
y = -(p*a^6+q*b^6)*b^2*(2*p*a^6-q*b^6)
z = b*(p*a^6+q*b^6)
w = (p*a^6+q*b^6)*a^2*(-2*q*b^6+p*a^6)

Especially, set (p,q,b)=(1,1,1) we get
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.


Case. (m,n)=(6,9)

x = a^2*(p*a^18+q*b^18)^2
y = -b^6*(2*p*a^18-q*b^18)*(p*a^18+q*b^18)^5
z = b^3*(p*a^18+q*b^18)^3
w = a^6*(p*a^18-2*q*b^18)*(p*a^18+q*b^18)^5






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