It has been a busy two weeks all over the math community. Well, at least it seemed so to me. Some of my friends have defended their theses and need only to walk to receive their PhDs; I completed my topics examination, Brown's take on an oral examination; and I've given a trio of math talks.
Meanwhile, there have been developments in a relative of the Twin Primes conjecture, the Goldbach conjecture, and Open Access math journals.
1. Twin Primes Conjecture
The Twin Primes Conjecture states that there are infinitely many primes $p$ such that $p+2$ is also a prime, and falls in the the more general Polignac's Conjecture, which says that for any even $n$, there are infinitely many prime $p$ such that $p+n$ is also prime. This is another one of those problems that is easy to state but seems tremendously hard to solve. But recently, Dr. Yitang Zhang of the University of New Hampshire has submitted a paper to the Annals of Mathematics (one of the most respected and prestigious journals in the field). The paper is reputedly extremely clear (in contrast to other recent monumental papers in number theory, i.e. the phenomenally technical papers of Mochizuki on the ABC conjecture), and the word on the street is that it went through the entire review process in less than one month. At this time, there is no publicly available preprint, so I have not had a chance to look at the paper. But word is spreading that credible experts have already carefully reviewed the paper and found no serious flaws.
Dr. Zhang's paper proves that there are infinitely many primes that have a corresponding prime at most $70000000$ or so away. And thus in particular there is at least one number $k$ such that there are infinitely many primes such that both $p$ and $p+k$ are prime. I did not think that this was within the reach of current techniques. But it seems that Dr. Zhang built on top of the work of Goldston, Pintz, and Yildirim to get his result. Further, it seems that optimization of the result will occur and the difference will be brought way down from $70000000$. However, as indicated by Mark Lewko on MathOverflow, this proof will probably not extend naturally to a proof of the Twin Primes conjecture itself. Optimally, it might prove the $p$ and $p+16$ - primes conjecture (which is still amazing).
One should look out for his paper in an upcoming issue of the Annals.
2. Goldbach Conjecture
I feel strangely tied to the Goldbach Conjecture, as I get far more traffic, emails, and spam concerning my previous post on an erroneous proof of Goldbach than on any other topic I've written about. About a year ago, I wrote briefly about progress that Dr. Harald Helfgott had made towards the 3-Goldbach Conjecture. This conjecture states that every odd integer greater than five can be written as the sum of three primes. (This is another easy to state problem that is not at all easy to approach).
One week ago, Helfgott posted a preprint to the arxiv that claims to complete his previous work and prove 3-Goldbach. Further, he uses the circle method and good old L-functions, so I feel like I should read over it more closely to learn a few things as it's very close to my field. (Further still, he's a Brandeis alum, and now that my wife will be a grad student at Brandeis I suppose I should include it in my umbrella of self-association). While I cannot say that I read the paper, understood it, and affirm its correctness, I can say that the method seems right for the task (related to the 10th and most subtle of Scott Aaronson's list that I love to quote).
An interesting side bit to Helfgott's proof is that it only works for numbers larger than $10^{30}$ or so. Fortunately, he's also given a computer proof for numbers less than than on the arxiv, along with David Platt. $10^{30}$ is really, really, really big. Even that is a very slick bit.
3. FoM has opened
I care about open access. Fortunately, so do many of the big names. Two of the big attempts to create a good, strong set of open access math journals have just released their first articles. The Forum of Mathematics Sigma and Pi journals have each released a paper on algebraic and complex geometry. And they're completely open! I don't know what it takes for a journal to get off the ground, but I know that it starts with people reading its articles. So read up!
The two articles are
-ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIESLeave a comment
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Comments (2)
2013-05-21 valuevar
Hi -
10^30 just seemed to be a nice, round number. The proof works starting at about 10^28 or so with some minor tweaking - and you can go further down if you are willing to cut close to the bone (I prefer to give myself some margin for slips). D. Platt and I had already checked all odd numbers up to 8.8*10^30, so 10^30 seemed to be a good point at which to slice the pie.
2013-05-22 davidlowryduda
Thanks Harald!