This is probably a well-known problem. Given a set or multiset of natural numbers let us construct its "Euclides" closure: on each step we take all possible products $P_i$ of the elements in the set, and add to the set those of $P_i+1$ who are primes. The question is whether this closure is finite or not.

For example, if we take $\{1\}$ as the initial set, then we add $2=1+1$, then we add $3=2+1$, then we add $7=2\cdot 3+1$, then we add $2\cdot 3\cdot 7+1=43$ and that is all. All other products of the elements of $\{1,2,3,7,43\}$, increased by 1 are composite numbers.

But if we start with the multiset $\{2,2\}$ then computations are much harder. We add $3=2+1,5=2\cdot2+1, 7=2\cdot3+1,11=2\cdot5+1,13=2\cdot2\cdot3+1,23=2\cdot 11+1$ and many many others.

Is "Euclides'"closure of $\{2,2\}$ finite?

Equivalent question: does there exist a positive integer $n$ such that whenever $p-1$ divides $4n$ for a prime number $p$, $p$ itself divides $2 n$.

twiceor justonce? $\endgroup$ – Greg Martin Apr 13 '18 at 19:24