# Prime numbers of the form $P^x+C$

Let it be $$\mathbb{P}$$ the set of prime numbers. Let it be $$p\in\mathbb{P}$$ some prime number, and $$C$$ some fixed constant such that $$p+C\in\mathbb{P}$$. One might wonder if there exist infinitely many $$x\in\mathbb{N}$$ such that $$p^{x}+C\in\mathbb{P}$$.

For instance, taking $$p=2$$, and noticing that $$2+1\in\mathbb{P}$$, it might be wondered if there exist infinitely many $$x\in\mathbb{N}$$ such that $$2^{x}+1\in\mathbb{P}$$. Until today, this particular case remains unanswered, because if is known that $$2^{x}+1\in\mathbb{P}$$ only if $$x$$ is of the form $$x=2^n$$, but it is not known if there exist infinite values of $$x$$ of the form $$x=2^n$$ such that $$2^{x}+1\in\mathbb{P}$$ (that is, it is not know if there exist infinitely many Fermat primes).

I am interested in known general results regarding this problem (if any).