Nontrivial Diophantine Solutions

[e2m2a]

TL;DR

Existence of non-trivial positive integer solutions.

Given the diophantine equation: 2x^5 - y^3 = 1 Is there any way I can prove that the only positive integer solution for this equation is: x =1, y = 1?

 

[fresh_42]

Why do you ask and what have you tried so far?

We could write $2x^5=(y+1)(y^2-y+1)$ and then we get $2\,|\,(y+1)$ because $y^2-y+1$ is odd.

 

[e2m2a]

Sorry, I don't know what you mean by the notation 2|(y+1)?

 

[pbuk]

It means '2 is a factor of (y + 1)'. I'm not sure if @fresh42 meant to say anything more?