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^6 + az^5 = y^6 + aw^5 has a parametric solution.
     
     
2.Theorem
     
Diophantine equation x^6 + az^5 = y^6 + aw^5 has a parametric solution.

(x,y,z,w)=(pa(r^5-s^5)(p^6-q^6)^4, qa(r^5-s^5)(p^6-q^6)^4, -ra(r^5-s^5)(p^6-q^6)^5, -sa(r^5-s^5)(p^6-q^6)^5)

a,p,q,r,s are arbitrary.
 
Proof.

x^6 + az^5 = y^6 + aw^5....................................................(1)

Substitute x=pt, y=qt, z=rt, w=st to equation (1), we obtain

(p^6-q^6)t+(ar^5-as^5)=0...................................................(2)

Hence we obatin t = -a(r^5-s^5)/(p^6-q^6).

Finally, we obtain a parametric solution below.

(x,y,z,w)=(pa(r^5-s^5)(p^6-q^6)^4, qa(r^5-s^5)(p^6-q^6)^4, -ra(r^5-s^5)(p^6-q^6)^5, -sa(r^5-s^5)(p^6-q^6)^5)

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