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? 



Must a prime power of the form 16t^2+1 be a prime?
[Catalan's Conjecture](https://en.wikipedia.org/wiki/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, 
 $$2^3+1=3^2$$


