It seems that big mathematical advances are like buses – you wait ages for one, and then two come along at once. Also revealed yesterday was a proof of the odd Goldbach conjecture: that all odd numbers greater than 7 can be written as the sum of exactly three odd primes. The proof is contained in Major arcs for Goldbach’s theorem, a paper submitted to the arXiv by Harald Helfgott, who’s a mathematician at the École Normale Supérieure in Paris. This new paper completes the work started in Helfgott’s previous paper, Minor arcs for Golbach’s problem, published last year.
The strong Goldbach conjecture states that every even number can be written as the sum of two primes. This is still unproven, and remains one of the long-standing unproven results in number theory. Sadly, it’s the opinion of Terence Tao, among others, that the method used to prove the weak conjecture probably won’t work on the strong conjecture.
The paper: Major arcs for Goldbach’s theorem by Harald Helfgott
via Terry Tao on Google+
I’m hijacking Katie’s newly-instituted series of posts about who to follow on Twitter with a post about who to follow on Google+.
Google+ famously has almost nobody on it. If anyone knows the potential for really interesting exceptions to the word “almost”, it’s mathematicians, so by that mad logic there should be some really interesting mathematicians on Google+. As luck has it, there are! It seems that the unconstrained nature of Google+ posts gives mathematicians the space they need to express themselves usefully.
Here are a few mathsy people you might like to encircle on Google+.
A conversation about mathematics between the UK and USA from Pulse-Project.org. This week Samuel and Peter spoke about: Every odd integer larger than 1 is the sum of at most five primes; No pardon for Alan Turing; more super bowl math; Early results from the Met Office weather game; Trends in Race/Ethnicity and Gender Representation in the Mathematical Sciences; Wolfram|Alpha Pro; more on Elsevier boycott; & more.
Download or stream via pulse-project.org.
Subscribe via Math/Maths on iTunes or Math/Maths RSS feed.
Terence Tao has uploaded to the arXiv a paper “Every odd number greater than 1 is the sum of at most five primes“, submitted to Mathematics of Computation. He says this result is:
in the spirit of (though significantly weaker than) the even Goldbach conjecture (every even natural number is the sum of at most two primes) and odd Goldbach conjecture (every odd natural number greater than 1 is the sum of at most three primes). It also improves on a result of Ramaré that every even natural number is the sum of at most six primes. This result had previously also been established by Kaniecki under the additional assumption of the Riemann hypothesis, so one can view the main result here as an unconditional version of Kaniecki’s result.