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 x 2 x 3 x 5 and 1 x 2 x 2 x 3 x 5 and 1 x 1 x 2 x 2 x 3 x 5 ...)
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 very first theorems 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 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 1,000 numbers, there are only 10 powers of 2: 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. In fact, it’s pretty apparent that out of the first X numbers, just about (1/2)X will be even.
Primes, it turns out, are intermediate—more common than the powers of 2 but rarer than even numbers. Among the first X numbers, about X/log(X) are prime; this is the Prime Number Theorem, proven at the end of the 19th century by Hadamard and de la Vallée Poussin. This means, in particular, that prime numbers get less and less common as the numbers get bigger, though the decrease is very slow; a random number with 20 digits is half as likely to be prime as a random number with 10 digits.
Naturally, one imagines that the more common a certain type of number, the smaller the gaps between instances of that type of number. If you’re looking at an even number, you never have to travel farther than 2 numbers forward to encounter the next even; in fact, the gaps between the even numbers are always exactly of size 2. For the powers of 2, it’s a different story. The gaps between successive powers of 2 grow exponentially, and there are finitely many gaps of any given size; once you get past 16, for instance, you will never again see two powers of 2 separated by a gap of size 15 or less.
Those two problems are easy, but the question of gaps between consecutive primes is harder. It’s so hard that, even after Zhang’s breakthrough, it remains a mystery in many respects.
And yet we think we know what to expect, thanks to a remarkably fruitful point of view—we think of primes as random numbers. The reason the fruitfulness of this viewpoint is so remarkable is that the viewpoint is so very, very false. Primes are not random! Nothing about them is arbitrary or subject to chance. Quite the opposite—we take them as immutable features of the universe, and carve them on the golden records we shoot out into interstellar space to prove to the ETs that we’re no dopes.
If you start thinking really hard about what “random” really means, first you get a little nauseated, and a little after that you find you’re doing analytic philosophy. So let’s not go down that road.
Instead, take the mathematician’s path. The primes are not random, but it turns out that in many ways they act as if they were. For example, when you divide a random number by 3, the remainder is either 0, 1, or 2, and each case arises equally often. When you divide a big prime number by 3, the quotient can’t come out even; otherwise, the so-called prime would be divisible by 3, which would mean it wasn’t really a prime at all. But an old theorem of Dirichlet tells us that remainder 1 shows up about equally often as remainder 2, just as is the case for random numbers. So as far as “remainder modulo 3” goes, prime numbers, apart from not being multiples of 3, look random.
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