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I was wondering if this is even posible because I can not see an answer...

Given a subset $S\subset P$ from the set of prime numbers $P$. Let $S$ have an infinite number of elements. Is there any subset $S$ and a polynomial $Q(x) = ax + b$ on integer coefficients with $a \neq 0, b \neq 0$ such that it maps $Q\colon S \rightarrow S^*$ with $S^* \subset P$?

For example, if the twin primes conjecture is true then one can choose $Q(x) = x + 2$ and $S = \{x \colon x \in P,\, x + 2 \in P \}$ given that there are infinitely many twin primes.

I just wonder if there is a simpler provable answer.

EffingLoveMath
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Ismael
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  • Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed. – Shaun Oct 30 '24 at 12:36

1 Answers1

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Posting @SassatelliGiulio 's comment as an answer to this interesting question.

Yitang Zhang proved the bounded prime gap theorem: there is an integer $k$ such that there are infinitely many prime pairs that differ by $k$. So the proved (not just provable) answer to the question is "yes" for the linear function $x+k$.

Whether there's a "simpler provable answer" is an open question. I think it unlikely.

Ethan Bolker
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