For a quick introduction to the theory of prime numbers, see my earlier post.
A mathematician by the name of Yitang Zhang made the news last week, for taking a step towards the twin prime conjecture. This conjecture asserts that there are infinitely many pairs of prime numbers that differ by 2. We call such pairs of numbers twin primes.
For example, 3 and 5 are twin primes, as are 5 and 7, 11 and 13, 17 and 19, and so on. It is not currently known if you can infinitely many of these pairs, and the resolution of this problem is considered as one of the greatest and most difficult problems in number theory.
So what did Zhang do? Well, Zhang, building on the work of others particularly Goldston, Pintz and Yildirim, has shown that there are infinitely many pairs of primes which differ by at most 70,000,000.
Ok, so 70,000,000 doesn’t seem that close to 2. But seeing as it’s the first result of this kind involving a fixed finite number, it really is spectacular! Furthermore, there really is no intuitive reason that primes should appear this close together. Well, rather, it is not intuitive to me!