I´m preparing for math contests and found the following problem from this pdf.
Find all integers $a, b, c >1$ and all prime numbers $p, q, r$ which satisfy the equation
$p^a=q^b+r^c$
($a, b, c$ and $p, q, r$ does not have to be distinct).
I guess we can solve this problem by examine possible divisors, from which we conclude that $p$ is a divisor in both $q$ and $r$, or none of them. But since $p, q, r$ are all primes, the first case is only possible if $p=q=r=2$.
If $p=q=r=2$ we see that $a=k+1$ and $b=c=k$ for some integer $k$ satisfies the equation. Let now $p, q, r$ be distinct. Then, because of parity, one (and only one) of them must be 2. But I´ve not come further than that. Any suggestions?