Prime Clusters and the Structure of Structurelessness

PRIME CLUSTERS AND THE STRUCTURE OF STRUCTURELESSNESS

Michell’s critical insight, that clusters of stars might appear to our eye even if stars were randomly distributed around our field of vision, doesn’t apply only to the celestial sphere. This phenomenon was the hinge for the pilot episode of the math/cop drama Numb3rs.* A series of grisly attacks, marked by pins on the wall map at HQ, showed no clusters; ergo, a single cunning serial killer intentionally leaving space between victims, not an unconnected burst of psychos, was at work. It was somewhat contrived as a police story, but mathematically it was perfectly correct.

The appearance of clusters in random data offers insight even in situations where there is no real randomness at all, like the behavior of prime numbers. In 2013, Yitang “Tom” Zhang, a popular math lecturer 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 primes. Zhang had been a star student at Beijing University, but had never thrived after moving to the United States for his PhD in the 1980s. He hadn’t published a paper since 2001. At one point, he left academic math entirely to sell sandwiches at Subway, until a fellow former student from Beijing tracked him down and helped him get an untenured lectureship at UNH. To all outward appearances, he was washed up. So it came as a great surprise when he released a paper proving a theorem some of the biggest names in number theory had tried, and failed, to conquer.

But the fact that the conjecture is true came as no surprise at all. Mathematicians have a reputation of being no-B.S. 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 closely related twin primes conjecture, even though it remains unproven. Why?

Let’s start with what the two 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, and 13.

Every positive integer 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, because 60= 2 × 2 × 3 × 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 . . .) What about prime numbers themselves? They’re fine; a prime number, like 13, is the product of a single prime, 13 itself. And what about 1? We’ve excluded it from our list of primes, so how can it be a product of primes, each one of which is larger than 1? Simple: 1 is the product of no primes.

At this point I’m sometimes asked, “Why is the product of no primes 1, and not 0?” Here’s one slightly convoluted explanation: If you take the product of some set of primes, like 2 and 3, but then divide away the very primes you multiplied, you ought to be left with the product of nothing at all; and 6 divided by 6 is 1, not 0. (The sum of no numbers, on the other hand, is indeed 0.)

The primes are the atoms of number theory, the basic indivisible entities of which all numbers are made. As such, they’ve been the object of intense study ever since number theory started. One of the first theorems ever proved in number theory is that of Euclid, which tells us that the primes are infinite in number; we will never run out, no matter how far along the number line we let our minds range.

But mathematicians are greedy types, not inclined to be satisfied with a mere assertion of infinitude. After all, there’s infinite and then there’s infinite. There are infinitely many powers of 2, but they’re very rare. Among the first one thousand numbers, there are only ten of them:

1, 2, 4, 8, 16, 32, 64, 128, 256, and 512.

There are infinitely many even numbers, too, but they’re much more common: exactly 500 out of the first 1,000 numbers. In fact, it’s pretty apparent that out of the first N numbers, just about (1/2)N will be even.

Primes, it turns out, are intermediate—more common than the powers of 2 but rarer than even numbers. Among the first N numbers, about N/log N are prime; this is the Prime Number Theorem, proven at the end of the nineteenth century by the number theorists Jacques Hadamard and Charles-Jean de la Vallée Poussin.