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Open Season: Prime Numbers (Part 1)

In this short series of articles, I’m writing about mathematical questions we don’t know the answer to – which haven’t yet been proven or disproven. This is the third article in the series, and across two parts will discuss various open conjectures relating to prime numbers.

I don’t think it’s too much of an overstatement to say that prime numbers are the building blocks of numbers. They’re the atoms of maths. They are the beginning of all number theory. I’ll stop there, before I turn into Marcus Du Sautoy, but I do think they’re pretty cool numbers. They crop up in a lot of places in maths, they’re used for all kinds of cool spy-type things including RSA encryption, and even cicadas have got in on the act (depending on who you believe).

On equivalent forms of the weak Goldbach conjecture

Harald Helfgott has announced a proof of the odd Goldbach conjecture (also known as the ternary or weak Goldbach conjecture). This is big news. Like a good maths newshound, Christian Perfect promptly wrote this up for The Aperiodical as “All odd integers greater than 7 are the sum of three odd primes!

Wait, though, there’s a problem. As Relinde Jurrius pointed out on Twitter, the formulation used in the paper abstract was not quite the same.

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $N$ greater than $5$ is the sum of three primes. The present paper proves this conjecture.

The version Christian used makes the assertion using odd primes, whereas the paper abstract only claims “the sum of three primes”. The latter version includes $7$ because $7$ can be written as the sum of three primes, but not odd ones ($7 = 3+2+2$). Certainly, you can see both statements of the weak Goldbach conjecture used (for example, here’s the $\gt 5$ version and here’s the $\gt 7$ version). Are they equivalent?

All odd integers greater than 7 are the sum of three odd primes!

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+