$a,b \in \mathbb N $ , $b$ odd $\implies$ $ \dfrac{2a^2-1}{b^2+2} \notin \mathbb Z $? If $a,b$ are positive integers and $b$ is odd , then is it ever possible that $ \dfrac{2a^2-1}{b^2+2} $ is an integer ?

Since $b$ is odd, $b^2 + 2\equiv 3\pmod{8}$. Thus $2$ is not a quadratic residue mod $(b^2 + 2)$. Hence $2a^2\not\equiv 1\pmod{b^2 + 2}$, from which the result follows.
