Busy day in analytic number theory; Harald Helfgott has complemented his previous paper http://arxiv.org/abs/1205.5252 (obtaining minor arc estimates for the odd Goldbach problem) with major arc estimates, thus finally obtaining an unconditional proof of the odd Goldbach conjecture that every odd number greater than five is the sum of three primes. (This improves upon a result of mine from last year http://terrytao.wordpress.com/2012/02/01/every-odd-integer-larger-than-1-is-the-sum-of-at-most-five-primes/ showing that such numbers are the sum of five or fewer primes, though at the cost of a significantly lengthier argument.) As with virtually all successful partial results on the Goldbach problem, the argument proceeds by the Hardy-Littlewood-Vinogradov circle method; the challenge is to make all the estimates completely effective and to optimise all parameters (which, among other things, requires a certain amount of computer-assisted computation).
Why is this interesting? Current cryptography is built on prime numbers. It “could” mean that there actually is more order in chaos than ever thought possible before, thus paving the way for extremely easy decryption of very sensitive data.