Question

\[5^x+7^y=z^2\] do positive integer solutions exist, for x,y,k

Answer 1

only solutions that can happen are when x or y is odd, they can't both be even.

Answer 2

okay so what else?
the theorem also states that x,y,k must share only 1 common prime factor?

Answer 3

I can give some hints!

Answer 4

Oh there's probably some way to solve this problem I'm just saying using Fermat's Last Theorem.

Beal Conjecture is what I mentioned in chat... it's unproven.

We can probably do something with quadratic reciprocity or residues or whatever though.

Answer 5

thats okay i just wanted to know this bealstuff too

Answer 6

to solve this i was gonna use something like the relatioship between 5 and 7s

Answer 7

http://norvig.com/beal.html