TITLE: Powers of a prime as one more than the square of an integer...
QUESTION [6 upvotes]: Given a fixed prime $p$, are there finitely many positive integers $k$ such that $p^k = n^2 +1$ for some $n$?

REPLY [4 votes]: The more general equation $n^2+1 = y^k$ for any positive integers $n$ and $y$ was proven to have no solutions for $k\geq 3$ by Lebesgue in 1850.  Obviously, there are no solutions for $k=2$ since the smallest difference between positive perfect squares is 3.  
See Bugeaud for complete details (including generalization to $n^2+D = y^k$).