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 ...).
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.
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 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.
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.
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.
Among the first N numbers, about N/log N of them are primes. If these were distributed randomly, each number n would have a 1/log N chance of being prime. The chance that n and n+2 are both prime should thus be about (1/log N)^2. So how many pairs of primes separated by 2 should we expect to see? There are about N pairs (n, n+2) in the range of interest, and each one has a (1/log N)^2 chance of being a twin prime, so one should expect to find about N/(log N)^2 twin primes in the interval.
A more refined analysis taking these into account suggests that the number of twin primes should in fact be about 32 percent greater than N/(log N)^2. This better approximation gives a prediction that the number of twin primes less than a quadrillion should be about 1.1 trillion; the actual figure is 1,177,209,242,304. That’s a lot of twin primes.
Despite the apparent simplicity of the bounded gaps conjecture, Zhang’s proof requires some of the deepest theorems of modern mathematics, like Pierre Deligne’s results relating averages of number-theoretic functions with the geometry of high-dimensional spaces.
