A Big Week in Number Theory

I know very little about number theory, and definitely do not follow progress in the field very closely. But two results were announced yesterday and today whose significance is not lost even to an outsider like me. They are both fundamental results about prime numbers and deal with questions that have been open for centuries.

The first result concerns the phenomenon of prime twins: although the prime numbers get farther apart as they get bigger, there still seem to be occasional pairs of primes that are close together; many are known, even some huge ones, that are two units apart. Now Yitang Zhang has shown that there are infinitely many primes that are no farther apart than 70 million. Although 70 million is much larger than two, this means that the phenomenon of twin primes does in a sense go on forever, since almost all primes are far larger than 70 million.

The second result, due to H. A. Helfgott, establishes the “ternary Goldbach conjecture”: every odd integer greater than five is the sum of three primes. Any result that reveals some structure to the prime numbers is interesting and important, since they in some sense appear to be random and unpredictable. For instance, there is no formula or deterministic procedure for discovering them, but they must be searched for in a kind of trial-and-error process, informed by whatever knowledge of their structure and distribution that we might possess.