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^8 + az^7 = y^8 + aw^7 has a parametric solution.
     
     
2.Theorem
     
Diophantine equation x^8 + az^7 = y^8 + aw^7 has two parametric solutions.

(x,y,z,w)=( pa(-r^7+s^7)(-q^8+p^8)^6, qa(-r^7+s^7)(-q^8+p^8)^6, ra(-r^7+s^7)(-q^8+p^8)^7, sa(-r^7+s^7)(-q^8+p^8)^7 )
a,p,q,r,s are arbitrary.

(x,y,z,w)=( p(-r^7+s^7), q(-r^7+s^7), r(-r^7+s^7), s(-r^7+s^7) )
a = p^8-q^8
p,q,r,s are arbitrary.

 
Proof.

x^8 + az^7 = y^8 + aw^7....................................................(1)

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

(-q^8+p^8)t = a(-r^7+s^7) = 0..............................................(2)

Hence we obatin t = a(-r^7+s^7)/(-q^8+p^8).

Finally, we obtain a parametric solution below.

(x,y,z,w)=( pa(-r^7+s^7)(-q^8+p^8)^6, qa(-r^7+s^7)(-q^8+p^8)^6, ra(-r^7+s^7)(-q^8+p^8)^7, sa(-r^7+s^7)(-q^8+p^8)^7 )

Let a = p^8-q^8, we obtain another parametric solution.
(x,y,z,w)=( p(-r^7+s^7), q(-r^7+s^7), r(-r^7+s^7), s(-r^7+s^7) )

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