Prime windows

Barry Cipra is one of the journalists covering the Heidelberg Laureate Forum. I’ve read his columns, especially in SIAM News, for years. It was great to meet him in person.

He and I were talking about the recent progress toward the twin prime conjecture and he had a way of describing the problem that I really liked.

The twin prime conjecture speculates that there are infinitely many pairs of primes that are two apart, like 3 and 5 or 11 and 13. The conjecture is still not proved, but earlier this year Yitang Zhang proved that there are infinitely many primes no more than 70,000,000 apart. That may not sound like a great result, but there was no finite upper bound before, and while 70,000,000 is a lot bigger than 2, it’s a lot less than infinity!

The people working on this problem speak of “gaps.” The gap first given by Zhang was 70,000,000. Over time the gap has fallen as mathematicians have gone over Zhang’s paper, making the estimates tighter. You can track the progress here.

The term “gap” bothered me. It doesn’t seem like the right term, since the gaps between the primes can be arbitrarily big. Barry told me he likes to use the term window instead. Zhang proved that if you take a window of length 70,000,000 and move it along the integers, you will infinitely often find a pair of primes inside your window. People have made improvements, shrinking the size of the window to something on the order of thousands. The twin prime conjecture says the window could be of length 2. (To be precise, the window at position n of size w is the closed interval [n, n+w].)

By the way, I said above that you can find arbitrarily large gaps in the primes. Here’s a simple proof. There are no primes between N! +2 and N! + N because N! + 2 is divisible by 2, N!+3 is divisible by 3, on up to N! + N which is divisible by N. You can make as long a prime-free interval as you like by taking N large enough.

I asked Barry about one more thing. Terence Tao said months ago that the current effort to shrink the prime window could not go all the way down to 2. That is, it could not prove the twin prime conjecture, but it could come up with improved partial results. How could Tao say that with confidence? Barry explained that the refinements to Zhang’s proof have been improving upper bounds on some functions. These same functions have known lower bounds, and so these lower bounds give a limit to how much the upper bounds can be reduced. The twin prime conjecture might be true, but a different approach would be necessary to prove it.

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  1. Great read John! I shall just add a couple of interesting (I hope) comments, for the study of large gaps actually holds some grand challenges.

    Indeed, when we are at N!, one can construct a gap of size about N. Interestingly, N is actually quite small compared to N!.

    It is known that when you are looking at numbers of size close to, say, n, the average gap between the primes is about log(n). The twin prime conjecture asserts that the gap (close to any n) can be as small as 2, but it is unknown as to how large the gap may be. Arbitrarily large, sure, but in comparison to n there is still much to explore.

    The Riemann Hypothesis would give that the maximum prime gap is at most the square root of n. I think it has been conjectured that the maximum gap is log^2(n).

    There’s also an Erdos prize of $10 000 available for progress on a related problem regarding large gaps.

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