First a warm-up. Let $\ V\ $ be an arbitrary set of odd natural numbers. Let $\ S(V)\ $ be the generated multiplicative semi-group. What are the necessary and/or sufficient conditions on $\ V\ $ for the property: $\ \exists_{x\ y\in S(V)}\ y-x=2\ $?

Now real questions, all of them open to me. Let $\ \mathbb P\ $ be the set of all primes. The following conjecture seems easier to proof than the twin primes conjecture(?):

**Q1.** Let real $\ a > 2\ $ be arbitrary. Let $\ V := [a;\infty)\cap\mathbb P.\ $ Then there exist $\ x\ y\in S(V)\ $ such that $\ y-x=2$.

Statement Q1 would hold if there were infinitely many twin primes. Can we obtain a similar result without any help from twins? I.e.

**Q2.** Let $\ p-2 < p < q < q+2\ $ be four primes. Let $\ V := [p;q]\cap\mathbb P.\ $ Is it true that for all such $\ p\ $ but a finite number, there exist $\ x\ y\in S(V)\ $ such that $\ y-x=2\ $?

A similar question can be asked when there only finitely many twin primes. Actually, we already have it. It is enough to replace $\ a\ $ of Q1 by the largest twin prime.

Finally, these questions can be posed in terms of numerical estimates:

**Q3.** For every natural $\ a>2\ $ compute or estimate the smallest natural $\ b:=b(a)\ $ such that there exist $\ x\ y\in S(V)\ $ such that $\ y-x=2\ $ for $\ V := [a;b]\cap S(\mathbb P)$.

Of course $\ b\ $ must be prime.

The above topic is a bit related to my earlier question: *Prime residua races and two views on primes*.