Hi, a friend of mine gave me a math problem which I've spent hours trying to find different methods to solve. But none of them work and I'm now out of ideas. The problem goes like this: So for example, ##17^2 = 1^2 + 2(12^2) ⇒ 17 = 3^2 + 2(2^2) ## or ## 3^2 = 1^2 + 2(2^2) ⇒ 3 = 1^2 + 2(1^2)## I also realized that this statement that we have to prove might also hold for all non-prime numbers as well. I found several examples that show that this is true: ##9^2 = 3^2 + 2(6^2) ⇒ 9 = 1^2 + 2(2^2)## or ## 12^2 = 4^2 + 2(8^2) ⇒ 12 = 2^2 + 2(2^2) ## So perhaps the information that 'p' is a prime plays no role whatsoever in the final proof. However, I'm not really sure about this. I've tried many methods so far, from indirect proofs (contradiction & contrapostive) to direct proofs, but none of them work. I've also looked at modular arithmetic, but that's getting nowhere. I'm really desperate for help at this moment. Any inputs from you guys are greatly appreciated, cuz my friend and I have been stuck for weeks on this question! Thanks a lot in advance!