# Nt.number Theory – On Diophantine equation \$ 3 ^ a + b ^ 2 = 5 ^ {c} \$, with \$ a, b, c in mathbb {N} \$

Consider the Diophantine equation
$$3 ^ a + b ^ 2 = 5 ^ {c},$$
with $$a, b, c in mathbb {N}$$.

My question: $$a, b, c$$ must be an even number?

Until now, I only understand that $$b$$ is an even number. To show this,
I suppose $$b = 2 b + 1$$. Note that $$5 ^ {c} – 3 ^ {a} = (2 c + 1) – (2 a + 1) = 2 (c – a)$$ (even number). So, $$b$$ can not be an odd number. On the other hand, I have not seen a basic way to prove that $$a, c$$ has the same property.