Number theory is chock-full of easily stated problems that are very difficult to solve. One such is the twin primes conjecture, which asserts simply that there are infinitely many twin primes. I’ll assume you know what a prime number is. Twin primes are primes that differ by exactly two, such as 3 and 5, 5 and 7, 11 and 13, 17 and 19, or 101 and 103.
Of course, everyone knows there are infinitely many primes. If you have not seen it before, here’s one easy way to prove that fact. We assume for a contradiction that there are only finitely many primes, let’s say k of them. Then we can make a list:
of all the primes. But now we can define a number N by multiplying all of these numbers together and adding one. That is, we have:
But we know that N must be divisible by some prime number, since every number has at least one prime factor. (Of course, the number N might itself be prime, but that’s OK, since every number is a factor of itself.)
We now ask, what prime number divides N? It cannot be any of the primes on our list, since N leaves a remainder of 1 when divided by any of those primes. The only possible conclusion is that there must be a prime number that is not on the list. This contradicts our assumption that our list was complete. It follows that no finite list can possibly contain all the primes, and therefore there are infinitely many prime numbers.
There is also a clever trick for showing that there are arbitrarily long sequences of non-prime numbers. Recall that if n is a positive integer, then by n!, read “n factorial”, we mean the product of all the integers from 1 to n. For example, we have
Notice that 6! is a multiple of the numbers 2, 3, 4, 5 and 6. I can use this fact to produce a sequence of five consecutive non-prime numbers, namely 722, 723, 724, 725 and 726. 722 is a multiple of 2, 723 is a multiple of 3, 724 is a multiple of 4, 725 is a multiple of 5 and 726 is a multiple of 6. Do you see the point? If you want n consecutive non-prime numbers, just consider the sequence:
The first number is divisible by 2, the next is divisible by 3, the next is divisible by 4, and so on until the last number, which is a multiple of n+1. Clever!
So, there are infinitely prime numbers, but also arbitrarily long gaps between primes. There is also the prime number theorem, which tells us that for very large values of x, the number of primes smaller than x is closely approximated by
(A more precise statement of the theorem can be found here.)
None of this really tells us much about twin primes, alas. But we do now know that there is some finite number with the property that there are infinitely many pairs of primes by differ by no more than that number:
On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.
Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.
Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.
“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”
What is that finite number? According to the conjecture, that number should be two. Here’s what Zhang achieved:
His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N. No matter how far you go into the deserts of the truly gargantuan prime numbers — no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million.
70 million might seem pretty far from 2. What you have to remember, though, is that the difference between 2 and 70 million pales in comparison to the difference between 70 million and infinity. So Zhang’s theorem is definitely a big deal!