# Sergei numbers : even integers n being a prime gap at least n times

Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number theory) numbers as even integers $n$ such that $n$ can be written in at least $n$ ways as the difference between consecutive odd primes. For example $2=13-11=7-5$ is a Sergei number.

Are there provably infinitely many Sergei numbers ? What about the best reachable lower bound of their density among even numbers ?

• This is likely to be true, with estimates coming from the Prime Number Theorem. In particular, if there are only finitely many such numbers, I can imagine statistics on gaps between primes skewed enough to contradict PNT. Gerhard "Need Lots Of Small Gaps" Paseman, 2018.06.28. – Gerhard Paseman Jun 28 '18 at 20:05
• Someone should mention that the standard conjectures on prime pairs imply that every even number has this property. (The prime pairs conjecture doesn't directly imply that such primes must be consecutive, but standard sieve arguments show that almost all occurrences of any fixed even gap occur between consecutive primes.) – Greg Martin Jun 29 '18 at 0:28

In fact, even stronger fact is true: every Polignac number is a Sergei number and therefore there is a constant $C$ such that every interval of the form $[M,M+C]$ contains at least one Sergei number. The corresponding result for Polignac numbers is proved in this article, Theorem 2.