Find all positive integer solutions for the following equation $5^a + 4^b = 3^c$: 
Find all positive integer solutions for the following equation:
$$5^a + 4^b = 3^c$$

My first guess would be to study the equation in mod, but I tried modulo 3, 4, 5, and 9 and I can't find anything.

Hint: $$1^a + 0 \equiv _4 (-1)^c \implies c \equiv_2 0$$
so $c=2d$, $d\in \mathbb{N}$.
So $$ 5^a = (3^d-2^b)(3^d+2^b)$$
so $3^d-2^b = 5^x$ and $3^d+2^b = 5^y$. Now add/substract both equations...


...we see that $x=0$. Now if $b>1$ then we have $$(-1)^d -0 \equiv _41 $$ so $d$ is even. So we have $$(3^m-1)(3^m+1)= 2^b$$ Now $(3^m-1)$ and $(3^m+1)$ are two consecutive even numbers so exactly one is divisible by $4$ so the other number is exactly $2$.

