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.