Find all positive integers $a$ for which there exists some positive integer $b$ s.t. $(2^a-1)\mid(b^2+9)$

Question

Find all positive integers $a$ for which there exists some positive integer $b$ s.t.

$(2^a-1)\mid(b^2+9)$

Answer 1

Since this is homework, I'll limit myself to some hints.

First, handle the cases $a=1$ and $a=2$.

Then, for $a\gt2$, consider prime divisors congruent $3$ mod $4$ of $2^a-1$ and of $b^2+9$.