Mathematicians Team Up To Close the Prime Gap 194
Hugh Pickens DOT Com writes "On May 13, an obscure mathematician garnered worldwide attention and accolades from the mathematics community for settling a long-standing open question about prime numbers. Yitang Zhang showed that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes separated by at most 70 million. His finding was the first time anyone had managed to put a finite bound on the gaps between prime numbers, representing a major leap toward proving the centuries-old twin primes conjecture, which posits that there are infinitely many pairs of primes separated by only two (such as 11 and 13). Now Erica Klarreich reports at Quanta Magazine that other mathematicians quickly realized that it should be possible to push this separation bound quite a bit lower. By the end of May, mathematicians had uncovered simple tweaks to Zhang's argument that brought the bound below 60 million. Then Terence Tao, a winner of the Fields Medal, mathematics' highest honor, created a 'Polymath project,' an open, online collaboration to improve the bound that attracted dozens of participants. By July 27, the team had succeeded in reducing the proven bound on prime gaps from 70 million to 4,680. Now James Maynard has upped the ante by presenting an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration's techniques with Maynard's approach to push this bound even lower. Zhang's work and, to a lesser degree, Maynard's fits the archetype of the solitary mathematical genius, working for years in the proverbial garret until he is ready to dazzle the world with a great discovery. The Polymath project couldn't be more different — fast and furious, massively collaborative, fueled by the instant gratification of setting a new world record. 'It's important to have people who are willing to work in isolation and buck the conventional wisdom,' says Tao. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."
regarding collaborative efforts (Score:5, Insightful)
sometimes its better to go it alone, then come back to the group with your results so that someone else may profit from them.
sometimes its better to be a part a group in order to establish your ideas and discuss, then go it alone when the group holds you back.
Re:Nice work (Score:5, Insightful)
Re:Summary (Score:5, Insightful)
Was it just me or did anyone else have a hard time following that summary? At first I thought it was Yitang Zhang who settled "a long-standing open question". But the first sentence is actually talking about the eight - James Maynard.
No. Before May 2013 there was no proof on an infinite pair of primes being a finite bound apart.
- May 2013: Zhang, bound 70 million
- End of May 2013: Others, bound <60 million
- July 2013: Terence Tao & Polymath project: bound 4680
- Now: James Maynard, bound 600
- Twin conjencture: still unproven, bound 2
So the "big" discovery was Zhang, for managing to put a bound on it in the first place. The rest are improvements on that proof, but not very fundamental ones. Proving the twin conjencture would be huge, but nobody's done that yet. The Polymath project and probably many others are working on it. The conjencture is almost certainly true, but notoriously hard to prove. Probably the easiest "feel" to get for it is the Sieve of Eratosthenes, make a long list of odd numbers then strike out the multiples of primes. Once you strike out the 3s it'll be obvious you don't get triplets since 3, 9, 15, 21, 27 and so on are all multiples of 3 so the "candidates" are (5,7) (11,13), (17,19), (23,25) and so on. As you add more primes like 25 = 5*5 it'll get fewer and fewer pairs but they keep occuring rather randomly. It feels like that with infinite primes they'll randomly end up being next to each other an infinite number of times, but proving it is another matter. For example if you take the Fibonacci sequence (1,1,2,3,5,8,13,21...) it's obvious it's an infinite series but the distance between numbers also grows to infinity. Not so with primes, by these proofs.
Re:Summary (Score:5, Insightful)
So in summary, if a pair of primes is defined by one following the other, it was theorized that we would find an infinite number of such pairs separated by 2. Various people have proven that gap to be from 70m, 60m, 4680, and now 600. Thank you James Maynard.
Here's what it real means: There were conjectures, one of them famous, which stated:
...
There are infinitely many pairs (p, p+2) of consecutive primes.
There are infinitely many pairs (p, p+4) of consecutive primes.
There are infinitely many pairs (p, p+6) of consecutive primes.
There are infinitely many pairs (p, p+600) of consecutive primes.
It is now proven that at least one of these conjectures is true.