![]() I have always felt that that project, despite not solving the problem, was a distinct success, because by the end of it I, and I was not alone, understood the problem far better and in a very different way. The most obviously successful polymath project was polymath8, which aimed to bring down the size of the gap in Zhang’s prime-gaps result, but it could be argued that success for that project was guaranteed in advance: it was obvious that the gap could be reduced, and the only question was how far.Īctually, that last argument is not very convincing, since a lot more came out of polymath8 than just a tightening up of the individual steps of Zhang’s argument. I started polymath5, with the aim of solving the Erdős discrepancy problem (after this problem was chosen by a vote from a shortlist that I drew up), and although we had some interesting ideas, we did not solve the problem. Terence Tao started polymath4, about finding a deterministic algorithm to output a prime between and, which did not find such an algorithm but did prove some partial results that were interesting enough to publish in an AMS journal called Mathematics of Computation. Gil Kalai started polymath3, on the polynomial Hirsch conjecture, but the problem was not solved. I started polymath2, about a Banach-space problem, which never really got off the ground. However, the subsequent experience made it look as though the first project had been rather lucky, and not necessarily a good indication of what the polymath approach will typically achieve. I will just make a few comments about what all this says about polymath projects in general.Īfter the success of the first polymath project, which found a purely combinatorial proof of the density Hales-Jewett theorem, there was an appetite to try something similar. I refer you to Terry’s posts for the mathematics. This post is therefore the final post of the polymath5 project. Two preprints covering (i) and (ii) are here and here: the one covering (i) has been submitted to Discrete Analysis. He has blogged about the solution in two posts, a first that shows how to reduce the problem to the Elliott conjecture in number theory, and a second that shows (i) that an averaged form of the conjecture is sufficient and (ii) that he can prove the averaged form. I imagine most people reading this will already have heard that Terence Tao has solved the Erdős discrepancy problem.
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