# Is there a theorem or conjecture that specifies a balanced prime must be multiple of 3# distance from neighboring primes?

I hope I have written this question in an answerable form. Basically, I am assuming this theorem or conjecture exists, but I can't find it from my reading / searching (I'm not a mathematician). What I know:

• A "balanced prime" is the middle prime number in a sequence of three consecutive primes in arithmetic progression. This would make the number "CPAP-3", and thus the minimum separation is 6 = 3# = 3 * 2.

• What I've noticed from just analyzing balanced primes (see my post on GooglePlus: I am too new of a user to post an image) is that not only is their minimum separation 6 = 3# = 3 * 2 (see the lowest magenta "line" in picture). But greater separations must be a multiple of 6 (by observation)

So, my question is: is there a named theorem or conjecture that describes the isolation of balanced primes must be a multiple of 3#? I'm not a mathematician, but I'm guessing this exists and I just can't find it.

I'm also very interested to know if there's a theorem or conjecture about how the maximum isolation of a balanced prime grows. It appears to grow faster than log(Pn).

Thanks!

PS: I labeled the graph with terms like "prime loneliness" before I was pointed to better terminology. "swiss primes" are actually "balanced primes."

EDIT (again): From comments: this is a proven (and easily explained) theorem (much more elementary than the Green-Tao theorem), which covers the more general case of primes in arithmetic progression (AP-K; not necessarily consecutive primes). Thank you to all for pointing me to the obvious! :) • The article you gave a link to mentions early on, with reference, a more general result about $k$ primes in AP. – André Nicolas May 19 '12 at 17:50
• 3,5,7 appears to be a sequence which has difference which is not a multiple of 3. – Mark Bennet May 19 '12 at 17:51
• @MarkBennet yes, 3,5,7 is an exception, but I think that would be covered in the theorem for minimum separation of a balanced prime (besides 5, all others have a minimum separation of 3#) – Steve Koch May 19 '12 at 19:02
• @AndréNicolas Thank you! I had read that, but didn't properly realize that CPAP-3 is a special case of AP-3. And apparently the Green-Tao theorem says: "If an AP-k does not begin with the prime k, then the common difference is a multiple of the primorial k# = 2·3·5·...·j, where j is the largest prime ≤ k." – Steve Koch May 19 '12 at 19:06
• @SteveKoch: That's not the Green-Tao Theorem, it is far more elementary. Green-Tao is the assertion that there are arbitrarily long (finite!) arithmetic sequences of primes. – André Nicolas May 19 '12 at 19:09

## 2 Answers

I don't think anyone has addressed here the question OP raised of the maximum isolation of a "balanced prime". It is conjectured (but I'm not sure whether it has been proved) that for any odd prime $p$ there are infinitely many three-term arithmetic progressions of primes with $p$ the smallest member. So, for example, there should be infinitely many primes $q$ such that $2q-3$ is prime, making $q$ a balanced prime with isolation $q-3$. In short, the isolation of a balanced prime can, conjecturally, be practically as big as the prime itself.

EDIT: On re-reading the question, I'm no longer sure what's being asked. Maybe OP means primes $p$ and positive integers $a$ such that $p\pm a$ are both prime and there are no other primes between $p-a$ and $p+a$. In that case, we're asking about gaps between successive primes, and there is a lot of literature on that; there are results, and widely-believed conjectures, and lots of room between the two. It's generally believed the gap between $p$ and the next prime maxes out at something like $(\log p)^2$. I see no reason why the isolation of "balanced" primes should be any different.

MORE EDIT: I notice that these primes are tabulated at the Online Encyclopedia of Integer Sequences. However I don't see anything there likely to aid in the estimation of maximum isolation.

• Thanks, @Gerry Myerson! I don't have reputation to upvote these answers, but thank you everyone for the links and help. – Steve Koch Jul 31 '12 at 13:28

If you have three numbers in arithmetic progression where the common difference is not divisible by 3, one of those three numbers will be divisible by 3. And in your case, that would mean it would not be prime (unless it were 3).