# Most Common Difference Between Two Consecutive Primes?

The question is as stated in the title. I was given this interesting problem by a friend of mine, but I don't know how to proceed with a solution. The immediate thought I had was that the most common difference is 2 since all primes are odd (except 2), but that seems trivial and silly. Any thoughts/hints/suggestions on how to find the most common difference? I suspect it has something to do with modular arithmetic but I'm not too sure.

Any and all help is appreciated :)

Thanks for reading,

A

Edit: To specify the term "most common," I mean to ask what is the most abundant or frequently occurring difference between two consecutive primes.

• Most common in what sense? Do you mean the average gap between primes? – Brandon Carter Apr 24 '15 at 2:35
• Most common as in the most occurring. I've made an edit to clarify. – A is for Ambition Apr 24 '15 at 2:41
• Read this article en.m.wikipedia.org/wiki/Prime_gap – marwalix Apr 24 '15 at 2:42
• Thanks marwalix! I didn't know about this subject. This gives me a good place to start. – A is for Ambition Apr 24 '15 at 2:43
• Most occurring is not normally a useful notion when there are infinitely many elements in the list. As far as I know, it is not even known whether a positive proportion of prime gaps are bounded. – Brandon Carter Apr 24 '15 at 2:59

## 1 Answer

From the PNT, the average prime gap around $n$ is $\sim\log n$, therefore the quantity you're interested in can't be bounded above. You can also take a glance on Marek Wolf's work about "jumping champions", which correspond to the most frequent prime gaps in a finite range, and which are conjectured to be the primorials, $2, 6, 30, 210...$. It is indeed related to modular arithmetic, since those numbers are in some sense "like $0$" as far as distance between primes is concerned. See also my question untitled "About Goldbach's conjecture" on Mathoverflow, or my blog ideasfornumbertheory.com for further insights, though expressed in an unrigorous fashion.

• I took a look at the very interesting article "Jumping Champions." So is the most common gap between successive primes just the largest primorial in that finite range? – A is for Ambition Apr 24 '15 at 20:58
• As I wrote above, it's a conjecture, no proof exists yet (as far as I know). – Sylvain Julien Apr 24 '15 at 21:04
• So the only definite answer to "what is the most common gap between successive primes" is that it's impossible to determine? – A is for Ambition Apr 24 '15 at 21:12
• The most reasonable answer would probably be "we don't know yet". – Sylvain Julien Apr 24 '15 at 21:48