For every $m \in \mathbb{N}$, does there exist some $n \in \mathbb{N}$ such that $m$ divides $n^2+1$?
This question came up as a small side question as part of some research I'm doing. I don't know a lot about number theory, and this looks like one of those questions that could either be really simple or totally beyond the limits of our current knowledge.
Best Answer
No.
$4$ cannot divide such an $n^2+1$, because modulo $4$, you have $n^2+1=1$ or $2$.