Is the following problem still open? Let $S$ be a nonempty 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?

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. 


Not claiming to be a real number theorist or anything of that sort...Just wondering, is one possible approach to Mark Sapir's strong conjecture (assuming an answer isn't already known) to calculate $$ \sum_{\substack{r,s < q\\ q rs+1}}\Lambda(r)\Lambda(s) $$ by $$ \int_0^1 \left(\sum_{r,s < q}\Lambda(r)\Lambda(s)e((rs+1)\alpha)\right)\left(\sum_{m <q}e(qm \alpha)\right)d\alpha $$ where $e(x) = e^{2 \pi i x}$ and then apply the circle method? If it can't be solved in this manner, perhaps it can still be made equivalent to a conjecture on some error terms. 


This was given as a problem in one of the MAA journals in the 1980's as a doublestarred problem (no solution provided). I met it as a challenge problem on an assignment in John Poland's (Carleton University) modern algebra course. I worked on it for a few months, observing as Mark Sapir does that such a set contains 2, 5, 11, etc. 


Dear Currie The below problem is still open: Do we have infinite prime number of the form $2p+1$, where $p$ is prime? One of interesting way to examine this problem is %Sandram$ table. For further information about Sandram table, you can see the book " Ingenuity in Mathematics" by Ross Hansberger Also, with some calculation and using the Jacobi symbol, you will find this problem is equivalent to this problem: The polynomial $n^2+n1$ generate infinite prime number, that it is open problem. You can find some further information about this question in the Richard Guy's book with name: " Open problem in number theory". So I think your problem is a kind of open problem. 

