[math-fun] Integers that are squares mod p for all p

I’m pretty sure that a positive integer n that is a square mod p^k for all prime powers p^k must be a square in Q (as a consequence of the local-to-global principle for quadratic forms), from which it follows that n must be a square in Z. But what if all we know is that n is a square mod p for every prime p? And what if we don’t know that n is positive? This question was raised by Alaric Stephen ( https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-1-alex-... ). Jim Propp