Consider a prime power q having the form 16t^2+1, where t is a positive integer. Numerical experiments show that when t is no greater than 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  ?

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?