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.

There are three solutions which can all be found by elementary means.
If $b$ is odd 
$$2^a+3\equiv 1 \bmod 4$$
Therefore $a=1$ and $b$ is odd.  
If $b>1$, then $2\equiv 5^c \bmod 9$ and $c\equiv 5 \bmod 6$ 
Therefore $2+3^b\equiv 5^c\equiv3 \bmod 7$ and $b\equiv 0 \bmod 6$, a contradiction. 
The only solution is $(a,b,c)=(1,1,1)$. 
If $b$ is even, $c$ is odd 
$$2^a+1\equiv 5 \bmod 8$$
Therefore $a=2$. 
If $b\ge 2$, then $4\equiv 5^c \bmod 9$ and $c\equiv 4 \bmod 6$, a contradiction. 
The only solution is $(a,b,c)=(2,0,1)$. 
If $b$ and $c$ are even 
Let $b=2B$ and $c=2C$. Then
$$2^a=5^{2C}-3^{2B}=(5^C-3^B)(5^C+3^B)$$
Therefore $5^C-3^B$ is also a (smaller) power of 2. 
A check of $(B,C)=(0,1)$ and $(1,1)$ yields the third solution $(a,b,c)=(4,2,2)$.
$(B,C)=(2,2)$ does not yield a further solution and we are finished.
