I'm taking the Putnam next fall, and decided to pick up a copy of Larson's problem solving book for practice. I'm having trouble, though, with one of the problems. It goes as follows. A well known theorem states that for a prime p>2, p=x^2+y^2 iff p is one more than a multiple of 4. Show: every prime one more than a multiple of 8 can be written in the form x^2 + 16y^2 :every prime five more than a multiple of 8 can be written in the form (2x+y)^2 + 4y^2 In all of these, x,y are integers. I think what might be confusing me is that I haven't taken a number theory course yet, so I don't know too much about mods. But if anyone can post solutions for these (as step-by-step as you can get please!!!) that would be very much appreciated. Thanks again.