Twin progress in number theory

Abstract

There are many jokes of the form "X's are like buses: you wait ages for one and then n show up at once." There appear to be many admissible values of {X,n}: {Ashes series,2}, {efficiency dividends, (m, where m > ∞)}. Normally, when X is progress on a problem in number theory, n is a non- negative integer, strictly less than unity. Therefore it was with ebullience that, on the same day, I read of the proof of a centuries-old problem, and of admirable, and completely unexpected, progress towards a millenia-old problem. This short note attempts to explain the two problems and to give a brief outline of the methods used to tackle them. Show more