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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).

Thanks in advance!

(This question is also posted in MSE, but got no answer and very few comments)

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