Try all numbers modulo 8 and see what is possible! (See Exercise 14.4.3.)
Already Fermat (unsurprisingly) claimed a partial converse to Fact 14.3.1. He stated that any prime number \(p\) which satisfies \(p\equiv 1\) or \(p\equiv 3\text{ (mod }8)\) could be written as a sum of a square and twice a square.
This time, Euler wasn’t the one who proved it! But you could almost imagine that by factoring