Jul 25 James Maynard of Oxford University won the Fields Medal this year for his work on prime numbers. Specifically, the Twin Primes Conjecture. Now I am drawn to mathematical work on numbers and primes like a horse to water (something like that). But what also struck me about Maynard is that his confession that the problems that particularly interest him are those that are easy to state, but have proved incredibly hard to solve.
The TPC is a good example. See what you make of my effort to explain, as best I can, Maynard's work on it (my math column last Friday July 22): The gap that wins the Fields, https://www.livemint.com/opinion/columns/the-gap-that-wins-the-fields-11658424609604.html And don't forget, I yearn for your reactions! cheers, dilip --- The gap that wins the Fields In a 2020 profile of James Maynard, one of the four mathematicians who won the Fields Medal this month, I found this sentence: "Maynard is drawn to questions about prime numbers that are simple enough to explain to a high school student but hard enough to stump mathematicians for centuries." He himself puts it this way in an accompanying video clip: "I'm always drawn to simple-to-state mathematical problems that have exceptionally difficult and interesting answers." That is probably the only mathematical connection I have with Maynard. It fascinates me that mathematics has many questions that even I, but certainly high school students, can understand and explain. Yet when it comes to proving them, they are hard, and plenty have not been proved. Take prime numbers. The famous Prime Number Theorem states that there is no end to them - that there is no such thing as a 'biggest" one, meaning they stretch to infinity. That they are infinite was first proved, and elegantly, by the ancient Greek mathematician Euclid. So that's something about primes that's both easily stated and actually proved. But consider twin primes, which are pairs of prime numbers that differ by two - like 11 and 13, or 137 and 139. Is there an infinity of them? Easily stated, again. But if primes are infinite, we don't know whether twin primes are. It seems so, because they keep appearing as we ratchet up through the numbers - for example, we know of a pair in which each has over 200,000 digits. Thus the Twin Primes Conjecture, that these twins are infinite in number. But "seems so" is not a proof. A logical proof has thus far resisted years of effort by a wide array of mathematical minds. Finding one remains one of the great unsolved problems in mathematics. But even if you don't find a solution to a hard mathematical problem, chiseling away at it produces worthwhile mathematical results. And who knows, maybe those results will win you a Fields Medal. That's what James Maynard's work on twin primes has done for him. "One of my favourite problems in mathematics," he says of the Conjecture in that clip. So let me give you a taste of his links to this favourite problem. In 2013, Yitang Zhang made mathematical waves with a result relevant to twin primes that, to folks outside mathematics, must seem almost laughable. He didn't prove that there's an infinity of primes that differ by two, but he did prove that there's an infinity of primes that differ by 70 million. Snicker if you like, but mathematicians like James Maynard recognized what this was: a major step towards proving the Twin Prime Conjecture. For they knew that 70 million and two are just numbers, and here represent just different-sized gaps between primes. The point is that we now know that the gap between primes is always finite - mathematically, that's a massive breakthrough. Shrinking that gap to two will be hard work, but now it's at least a nearly tangible goal. Plenty of brilliant mathematicians went at it hard, and within a year after Zhang's result, their collaborative effort (the so-called "Polymath Project") had used Zhang's techniques to reduce the gap to 246. Meanwhile, Maynard made progress towards finding a limit for the gap too, using different methods. In late 2013, he submitted a paper called "Small Gaps Between Primes" to the Annals of Mathematics ( https://arxiv.org/pdf/1311.4600.pdf). In it, he proved that the gap was at most 600. While acknowledging Zhang's breakthrough, he noted that his result "does not incorporate any of the technology used by Zhang ... the proof is essentially elementary, relying only on the Bombieri-Vinogradov Theorem." Never mind what the BVT is, but Maynard also commented: "600 ... is certainly not optimal. [With] further numerical calculations our method could [reduce it], and ... Zhang’s work and the refinements produced by the polymath project should be able to be combined with this method to reduce [it] further." Maynard's research actually promised even more. Still another conjecture about prime numbers is known as the Elliot-Halberstam Conjecture. Never mind what that is either. Like the Twin Primes Conjecture, the EHC is yet to be proved, though the BVT mentioned above proves one part of it (you might say "half" of it). But in his "Small Gaps" paper, Maynard showed that if we assume the EHC is true, the gap we are concerned with is just ... 12. Some months later, the polymath project had reduced that limit to just ... 6. That is, if the EHC is true, we know that there is an infinity of pairs of consecutive primes that differ by no more than 6. Naturally, this means they differ by either 2, 4, or 6 (they cannot differ by 1, 3 or 5 because then one of the pair would be even, thus not prime). So they are either twin primes (2), so-called cousin primes (4) or sexy primes (6). Oh yes, mathematicians call certain primes sexy. Don't ever imagine they are staid, buttoned-down folks. As you might guess, Maynard's work is far wider and more complex - maybe even sexier - than what I've hinted at here. He has built on the work of Bombieri and Vinogradov, Zhang and the polymath project, to produce important and interesting results about how primes are distributed among the numbers. He has proved there are infinitely many primes that don't contain a given digit, like 3, or 6. And yes, he has also addressed large gaps between primes. His 2016 paper of that name showed that "there exist pairs of consecutive primes ... whose difference is larger than" a certain lower limit. Still, Maynard does think that the small gap of 6 is insurmountable without some brand-new mathematical insight and tools. Which means that whoever comes up with those may just win a future Fields Medal. -- My book with Joy Ma: "The Deoliwallahs" Twitter: @DeathEndsFun Death Ends Fun: http://dcubed.blogspot.com -- You received this message because you are subscribed to the Google Groups "Dilip's essays" group. To unsubscribe from this group and stop receiving emails from it, send an email to dilips-essays+unsubscr...@googlegroups.com. To view this discussion on the web, visit https://groups.google.com/d/msgid/dilips-essays/CAEiMe8q2Wg6cb9gC%3DSagz5ZbijgEvt%2Bo5nbyd1xShv%3Ds9HA9oQ%40mail.gmail.com.
