Is the following problem still open? Let $S$ be a non-empty set of prime numbers such that whenever $p,q\in S$, all the prime factors of $pq+1$ are also elements of $S$. Is $S$ infinite?
[Math] Generalizing Euclid’s proof of the infinity of primes
nt.number-theoryprime numbers
Best Answer
The answer is most probably "It is always the set of all prime numbers or empty". Indeed, assume that $S$ is not empty. First of all, $S$ must contain $2$ (if it contains $p>2$, then $p^2+1$ is even). Then it also contains $2\cdot 2+1=5, 2\cdot 5+1=11, 7\mid 5\cdot 11+1, 3\mid 2\cdot 7+1,$ etc. It looks like then $S$ contains all prime numbers. I cannot prove it yet, but the proof should not be that difficult.
Update Here is a stronger conjecture. Let $S$ be the set of all primes from $2$ to $p>17$. Let $q$ be the next prime after $p$. Then $q$ divides $rs+1$ for some $r,s\in S$. This of course implies the statement above. I wonder if my conjecture is already known. I checked it for all $p<100,000$. I stop checking and will wait for an opinion of a real number theorist.