Can twice a perfect square be divisible by $q^{\frac{q+1}{2}} + 1$, where $q$ is a prime with $q \equiv 1 \pmod 4$? Can twice a perfect square be divisible by 
$$q^{\frac{q+1}{2}} + 1,$$
where $q$ is a prime with $q \equiv 1 \pmod 4$?

Try proving something "harder":
Theorem: Let $n$ be a positive integer. There exists a positive integer $k$ such that $n | 2k^2$

 k=n

