# 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! :)

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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

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.