A huge discovery about prime numbers—and what it means for the future of math.

By Jordan Ellenberg

The MacArthur Foundation announced Wednesday that Yitang Zhang has been awarded one of its 2014 fellowships, also known as genius grants. The foundation cited him for “probing with original insights into number theory.” Last year, Jordan Ellenberg explained that Zhang’s discovery about prime numbers points to a richer theory of randomness.

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.

The stereotype, outmoded though it is, is that new mathematical discoveries emerge from the minds of dewy young geniuses. But Zhang is over 50. What’s more, he hasn’t published a paper since 2001. Some of the world’s most prominent number theorists have been hammering on the bounded gaps problem for decades now, so the sudden resolution of the problem by a seemingly inactive mathematician far from the action at Harvard, Princeton, and Stanford came as a tremendous surprise.

But the fact that the conjecture is true was no surprise at all. Mathematicians have a reputation of being no-bullshit hard cases who don’t believe a thing until it’s locked down and proved. That’s not quite true. All of us believed the bounded gaps conjecture before Zhang’s big reveal, and we all believe the twin primes conjecture even though it remains unproven. Why?

Let’s start with what the conjectures say. The prime numbers are those numbers greater than 1 that aren’t multiples of any number smaller than themselves and greater than 1; so 7 is a prime, but 9 is not, because it’s divisible by 3. The first few primes are: 2, 3, 5, 7, 11, 13 …

Every positive number can be expressed in just one way as a product of prime numbers. For instance, 60 is made up of two 2s, one 3, and one 5. (This is why we don’t take 1 to be a prime, though some mathematicians have done so in the past; it breaks the uniqueness, because if 1 counts as prime, 60 could be written as 2 × 2 × 3 × 5 and 1 × 2 × 2 × 3 × 5 and 1 × 1 × 2 × 2 × 3 × 5 …)

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Nice story, Boris.

I followed a link on that page to Terrence Tao’s blog and then my head exploded :)

>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.<

Let’s just hope they’ll be able to translate these concepts into English.

Bubblecar said:

>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.<

Let’s just hope they’ll be able to translate these concepts into English.

It reads better in the original Klingon dialect

From “more”.

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?

Nothing of the kind. If you strew numbers at random, it’s very likely that some pairs will, by chance, land very close together.

It’s not hard to compute that, if prime numbers behaved like random numbers, you’d see precisely the behavior that Zhang demonstrated. Even more: You’d expect to see infinitely many pairs of primes that are separated by only 2, as the twin primes conjecture claims.

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Aha, so that’s what it’s about.

Cymek said:

Bubblecar said:

>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.<

Let’s just hope they’ll be able to translate these concepts into English.

It reads better in the original Klingon dialect

That paragraph nicely parallels a prediction of the future by Feynman many years ago. His prediction was that in future years we’d get a breakthough in understanding randomness, and may even come up with a new kind of randomness.