TITLE: How to solve $(2x^2-1)^2=2y^2 - 1$ in positive integers?
QUESTION [5 upvotes]: I encountered this question (posed by Fermat) in a letter from Fermat to Carcavi and was wondering what would be the best elementary way to solve it. 

Solve in positive integers$$(2x^2-1)^2=2y^2 - 1$$  

Any help will be appreciated.
Thanks in advance.

REPLY [2 votes]: For $y$ a prime, the simplest solution I've ever seen is this answer in another thread. I have no doubt that (very elementary) method could be adapted to solve the fully-general theorem. Another solution, given by Mahoney, I replicated here.