And I stand by the assertion that all primes other than 2 can be written as the difference of two consecutive squares. Primes that have this property are called Achernar primes. 
Cool, I understand you now. Yeah, no doubt, every prime other than 2 is of the form 4n+1 or 4n-1. My point was that Fermat didn’t prove that, that’s been known for thousands of years. I had a feeling you were referring in some way or another to sums of squares–that’s what Fermat proved (I don’t know whether his proof of this is known or not), that primes of the form 4n+1 can be written as the sum of two squares.
More generally, every prime, in fact, every positive integer, can be written as the sum of four squares (three is not enough). Fermat wrote that he had proven this, but it was never published (it’s commonly believed that he did, in fact, prove it, though). Lagrange was the first to publish a proof of this fact, in 1770.
This bear thinks that one is a prime number: