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on Erdős #1209 · number theory

The statement

Let $A=\{a_1<a_2<\cdots\}$ be a sequence of integers which tends to infinity sufficiently fast. If there is an $n$ such that all $n+a_k$ are primes then must there exist infinitely many such $n$ ?

What if we ask for $n+a_k$ to be squarefree instead of prime?

Are there $n$ such that $n+2^{2^k}$ is always a prime (or always squarefree, or infinitely often a prime, or infinitely often squarefree)?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.