What prime powers differ by one I thought this would be a hard problem but I found a link that seems to ask the answer to this question as a homework problem? Can somone help me out here, are there an infinite number of prime powers that differ by 1? or are there a finite number of them? If so which are they?

Catalan's Conjecture, a theorem since $2002$, shows that the only examples where the exponents are $\ge 2$ are $3^2-2^3$.
If we allow exponent equal to $1$, the answer is not known. Perhaps there are infinitely many Fermat primes, that is, primes of the form $2^n+1$ (it turns out that for $2^n+1$ to be prime, we need $n$ to have shape $2^k$). 
For many years, only five such primes have been known. There may not be others, or there may be finitely many others, or infinitely many. At the current time, we do not know.
