1.Introduction

A.S. Janfada and  A. Abbaspour[1] showed that equation x^6+6z^3=y^6+6w^3 has infinite non-trivial primitive integer solutions.

We show that x^5 + az^2 = y^5 + aw^2 has a parametric solution.
     
     
2.Theorem
     
Diophantine equation x^5 + az^2 = y^5 + aw^2 has a parametric solution.

(x,y,z,w)=(4pt^2a^2, 4qt^2a^2, 16(t^2q^5-t^2p^5+a)t^4a^4, 16t^4a^4(-t^2q^5+t^2p^5+a))
a,p,q,t are arbitrary.
 
Proof.

x^5 + az^2 = y^5 + aw^2....................................................(1)

Substitute x=p, y=q to equation (1), we obtain

a(z+w)(z-w) = q^5-p^5......................................................(2)

Hence we obatin (z,w) =( 1/2(t^2q^5-t^2p^5+a)/(ta), -1/2(-t^2q^5+t^2p^5+a)/(ta) ).

Finally, we obtain a parametric solution below.

(x,y,z,w)=(4pt^2a^2, 4qt^2a^2, 16(t^2q^5-t^2p^5+a)t^4a^4, 16t^4a^4(-t^2q^5+t^2p^5+a))

Q.E.D.



3.Reference

[1].A.S. Janfada and  A. Abbaspour, On Diophantine equations X^6+ 6Z^3 = Y^6± 6W^3,
    International Journal of Pure and Applied Mathematics, VOL:105, NO:4,2015