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^7 + az^6 = y^7 + aw^6 has a parametric solution.
     
     
2.Theorem
     
Diophantine equation x^7 + az^6 = y^7 + aw^6 has a parametric solution.

(x,y,z,w)=(-pa(-r^6+s^6)(-p^7+q^7)^5, -qa(-r^6+s^6)(-p^7+q^7)^5, ra(-r^6+s^6)(-p^7+q^7)^6, sa(-r^6+s^6)(-p^7+q^7)^6)

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

x^7 + az^6 = y^7 + aw^6....................................................(1)

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

(p^7-q^7)t+(ar^6-as^6)=0...................................................(2)

Hence we obatin t = -a(-r^6+s^6)/(-p^7+q^7).

Finally, we obtain a parametric solution below.

(x,y,z,w)=(-pa(-r^6+s^6)(-p^7+q^7)^5, -qa(-r^6+s^6)(-p^7+q^7)^5, ra(-r^6+s^6)(-p^7+q^7)^6, sa(-r^6+s^6)(-p^7+q^7)^6)

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