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^4 = y^5 + aw^4 has a parametric solution.
     
     
2.Theorem
     
Diophantine equation x^5 + az^4 = y^5 + aw^4 has a parametric solution.

(x,y,z,w)=(-pa(r^4-s^4)(p^5-q^5)^3, -qa(r^4-s^4)(p^5-q^5)^3, ra(r^4-s^4)(p^5-q^5)^4, sa(r^4-s^4)(p^5-q^5)^4)

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

x^5 + az^4 = y^5 + aw^4....................................................(1)

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

(p^5-q^5)t+(ar^4-as^4)=0...................................................(2)

Hence we obatin t = -a(r^4-s^4)/(p^5-q^5).

Finally, we obtain a parametric solution below.

(x,y,z,w)=(-pa(r^4-s^4)(p^5-q^5)^3, -qa(r^4-s^4)(p^5-q^5)^3, ra(r^4-s^4)(p^5-q^5)^4, sa(r^4-s^4)(p^5-q^5)^4)

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