Consider a prime power $q$ having the form $16t^2+1$, where $t$ is a positive integer. Numerical experiments show that when $t \leq 10^9$, each prime power $q$ with this form is indeed a prime.
In general, is it true that a prime power of this form must be a prime?
Catalan's Conjecture is a theorem (as of 2002):
With one exception, there are no solutions in positive integers of $$x^a+1=y^b$$ with $a,b \ge 2.$
The exception is, of course, $$2^3+1=3^2$$
There has been a comprehensive treatment on the solutions of Diophantine equation with the form $x^2+C=y^n$, where $x$ and $y$ are positive integers, $n \ge 3$ and $C$ is a positive integer. (J. H. E. Cohn, "The diophantine equation $x^2+C=y^n$", Acta Arithmetica,1993).
As a special case, when $C=1$, the equation $x^2+1=y^n$ with $n \ge 3$, has no solution.
