May 2013 turned out to be an interesting month for number theory, in that there’ve been big breakthroughs on two long-standing conjectures. Number theory is great fun in part because it’s got many conjectures that anybody can understand but which are really hard to prove. The one that’s gotten the most attention, at least from the web sites that I read which dip into popular mathematics, has been the one about twin primes.
It’s long been suspected that there are infinitely many pairs of “twin primes”, such as 5 and 7, or 11 and 13, or 29 and 31, separated by only two. It’s not proven that there are such, not yet. Yitang Zhang of Harvard has announced proof that there are infinitely many pairings of primes that are no more than 70,000,000 apart. This is admittedly not the tightest bound out there, but it’s better than what there was before. But while there are infinitely many primes — anyone can prove that — how many there are in any fixed-width range tends to decrease, and it would be imaginable to think that the distance between primes just keeps increasing, without bounds, the way that (say) each pair of successive powers of two is farther apart than the previous pair were. But it’s not so, and that’s neat to see.
Less publicized is a proof of Goldbach’s Odd Conjecture. Goldbach’s Conjecture is the famous one that every even number bigger than two can be written as the sum of two primes. An equivalent form would be to say that every whole number — even or odd — larger than five can be written as the sum of three primes. Goldbach’s Odd Conjecture cuts the problem by just saying that every odd whole number greater than five can be written as the sum of three primes. And it’s this which Harald Andres Helfgott claims to have a proof for. (He also claims to have a proof that every odd number greater than seven can be written as the sum of three odd primes, that is, that two isn’t needed for more than single-digit odd numbers.)