Bounded Gap Conjecture

“Last week, Yitang “Tom” Zhang, a popular math professor at the University of New Hampshire,  stunned the world of pure mathematics when he announced that he had proven the “bounded gaps” conjecture about the distribution of prime numbers—a crucial milestone on the way to the even more elusive twin primes conjecture, and a major achievement in itself.”

“What about the gaps between consecutive primes? You might think that, because prime numbers get rarer and rarer as numbers get bigger, that they also get farther and farther apart. On average, that’s indeed the case. But what Yitang Zhang just proved is that there are infinitely many pairs of primes that differ by at most 70, 000, 000. In other words, that the gap between one prime and the next is bounded by 70, 000, 000 infinitely often—thus, the “bounded gaps” conjecture.”

“On first glance, this might seem a miraculous phenomenon. If the primes are tending to be farther and farther apart, what’s causing there to be so many pairs that are close together? Is it some kind of prime gravity?”

“Building on the work of many predecessors, Zhang is able to show in a rather precise sense that the prime numbers look random in the first way we mentioned, concerning the remainders obtained after division by many different integers. From this (following a path laid out by Goldston, Pintz, and Yıldırım, the last people to make any progress on prime gaps) he can show that the prime numbers look random in a totally different sense, having to do with the sizes of the gaps between them. Random is random!”

“Zhang’s success (along with the work of Green and Tao) points to a prospect even more exciting than any individual result about primes—that we might, in the end, be on our way to developing a richer theory of randomness. How wonderfully paradoxical: What helps us break down the final mysteries about prime numbers may be new mathematical ideas that structure the concept of structurelessness itself.”

 

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