Consider the number $(2m)^{2^n}+1$ where both $m$ and $n$ are positive integers.
Can it be shown that for any given $n$, there exists an $m$ such that $(2m)^{2^n}+1$ is a prime number.
Edit: It can be shown that there exists an integer k such that (2m)2n can be written k2 for any integers m and n. If the conjecture in the question was true, then it would thus imply there are an infinite number of primes of the form k2+1. This is Landau's fourth problem which I believe still remains unsolved. Therefore I suspect this conjecture also cannot be proved at the current time