Finding all natural solutions to $2^a+5=b^2$ What are all possible natural number solutions $(a, b)$ to the equation

$$2^a+5=b^2 ?$$

The only solution I've found is 
$$2^2+5=3^2.$$
This may be the only solution but I don't have a proof.

For the first equation we can see that $b$ is odd, since $2^a+5=b^2$ and $2^a$ is always even. So rewriting the equation you have $2^a+4= (b-1)(b+1)$. Is clear that if $a=1$, it doesn't has solution so $a\geq 2$. So
$$4(2^{a-2}+1)= (b-1)(b+1)$$
But $b$ is odd, so one between $b-1$ and $b+1$ is divided by 2 and the other by 4 (they're two consecutive even numbers), so the right side is divided by 8, and the only way for the left side to be divisible by 8 is when $2^{a-2}+1$ is even, that is when $2^{a-2}$ is odd, that only happens when $2^{a-2}=1$, so $a-2=0$, that lets $a=2$, that is the only solution you found.
