Update 17/06/2013: The gap is down to 60,744. That’s a whole order of magnitude down from where it started!
When Yitang Zhang unexpectedly announced a proof that that there are infinitely many pairs of primes less than 70 million apart from each other – a step on the way to the twin primes conjecture – certain internet wags amused themselves and a minority of others with the question, “is it a bigger jump from infinity to 70 million, or from 70 million to 2?”.
Of course the answer is that it’s a really short distance from 70 million to 2, and here’s my evidence: the bound of 70 million has in just over three weeks been reduced to just a shade over 100,000.
There were two nice things about Yitang Zhang’s paper: he explains the techniques he used very well; and the bound of 70,000,000 that he announced could be whittled down fairly easily – the big achievement was showing that there is a bound at all, never mind how big it is. In fact, Zhang rounded his actual bound of 63,374,611 up to 70 million, because it doesn’t really matter.
But if there is wiggle room, people will inevitably wiggle. Straight away, mathematicians around the world set to work fine-tuning the bounds on the various quantities involved in Zhang’s proof. The core result is that if a set $S$ of $3.5 \times 10^6$ numbers is admissible, then there are infinitely many $n$ such that $n+S$ contains at least two primes. If the difference between the largest and the smallest elements of the admissible set is $k$, then you know that there are infinitely many prime numbers at most $k$ apart. So constructing a narrower admissible set is the name of the game.
As far as I can tell, Tim Trudgian was first to bid, with a bound of 59,874,594, which he called “a poor man’s improvement” in a note submitted to the arXiv. Soon after that, Scott Morrison of the Secret Blogging Seminar found a way to reduce it ever so slightly to 59,470,640.
The bound quickly tumbled much further than that, and Terry Tao initiated a new Polymath project to coordinate efforts. Participants are trying to optimise three constants:
- $H$ is a quantity such that there are infinitely many pairs of consecutive primes of distance at most $H$ apart. Would like to be as small as possible.
- $k_0$ is a quantity such that every admissible $k_0$-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible.
- $\varpi$ is a technical parameter related to a specialized form of the Elliott-Halberstam conjecture. Would like to be as large as possible. Improvements in $\varpi$ lead to improvements in $k_0$.
Records are being kept on the Polymath wiki, and progress is swift: just yesterday eighteen different values for $H$ were recorded, along with four values of $k_0$. As I write, they currently think that $H \leq 108,540$. Of course, in the rough and tumble of mathematicians trading arguments through comments pages it’s entirely possible that mistakes will go unnoticed, but it’s certainly fun to watch!
Interested followers should take a look at an online reading seminar conducted by Terry Tao which is going through the original paper.
Bounded gaps between primes by Yitang Zhang
A poor man’s improvement on Zhang’s result: there are infinitely many prime gaps less than 60 million by T.S. Trudgian
I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart by Scott Morrison
Polymath proposal: bounded gaps between primes by Terence Tao
Online reading seminar for Zhang’s “bounded gaps between primes” at Terry Tao’s blog
Bounded gaps between primes at the Polymath project wiki
Game of proofs boosts prime pair result by millions at New Scientist