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?
Here are some nice number facts and tricks you can try out on your friends. They will work without understanding how, but with a little investigation you should be able to figure out how each one works.
Update 14/05/2013: The seminar was successful: Zhang announced that his proof has already been refereed for the Annals, and everyone seems happy with it.
Hard Maths news now: there’s a rumour going round that Yitang (Tom) Zhang of the University of New Hampshire reckons he can prove that there are infinitely many different pairs of primes at most 70,000,000 apart.
The London Mathematical Society Popular Lectures for 2013 have been announced. Professor Ray Hill, University of Salford, will talk about ‘Mathematics in the Courtroom’, and Dr. Vicky Neale, University of Cambridge, will give a lecture on ‘Addictive Number Theory’.
One would be hard put to ﬁnd a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery — and more totally useless — than the perfect numbers.
— Martin Gardner
There are countless ways to classify integers. Happy, perfect, friendly, sociable, abundant, extravagant, cute, interesting, frugal, deficient, hungry, undulating, weird, vampire… the list goes on. But how useful are such classifications, beyond their inherent interestingness, and as a hook to get people into number theory?