<s>[INST] The positive integer solutions for $2^a+3^b=5^c$ What are the positive integer solutions to  the equation

$$2^a + 3^b = 5^c$$ 

Of course $(1,\space 1, \space 1)$ is a solution.
 [/INST] 
case : $b>2$ and $c>2$

A calculus with computer, modulo $N=2372625=5^3\times 19 \times 37 \times 3^3$, in $H=\mathbb Z/N\mathbb Z  $ give 
$\text{card}(<2>)=900$, $\text{card}(3^3\times <3>)=900$, $\text{card}(5^3\times<5>)=36$.
And $(<2>+(3^3\times <3>)) \cap 5^3\times <5>=\emptyset$
with $1\in <g>$ the subset of $H$ generated by $g$ with times ($\times$)
So if $b>2$ and $c>2$, there are not solutions.

case : $b\leq 2$ and $c>2$

Here we choose $N=5^3\times (2^{25}-1)$
so $(<2>+\{1,3,9\}) \cap 5^3\times <5>=\emptyset$
So if $b\leq 2$ and $c>2$ no solution.

case : $c \leq 2$ three solutions

Then there are only 3 solutions, not any more.
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