Twins move within reach

Agreeing to pay £50,000 for something worth £2 wouldn't win you any haggling competitions. In mathematics, however, a similar result can bring you international acclaim. Mathematics can be a lot like haggling: if you can't get the answer you want, you try to get as close as possible. In the case of the famous twin prime conjecture the answer you want is 2. Recent results claim to have managed just over 50,000, but they are causing quite a stir. That's because the gap between 2 and 50,000 is a whole lot smaller than the gap between 50,000 and infinity.

The twin prime conjecture involves gaps between prime numbers: those natural numbers that are only divisible by themselves and 1. The first few are 2, 3, 5, 7, 11, and 13. Over 2000 years ago the Greek mathematician Euclid proved that there are infinitely many of them but the question of how they are scattered among the other numbers continues to fox mathematicians. As you go up the number line the primes become more sparse, but every now again you hit upon pairs of primes that are two apart. The first ones are 3 and 5, 5 and 7, 11 and 13. Moving up the number line we find, for example, 641 and 643 and, later on, 857 and 859. The largest pair of twin primes found so far is $3756801695685 \times 2^{666669} + 1$ and $3756801695685 \times 2^{666669} - 1.$ These two numbers are 200,700 digits long and they were found using a computer search which took over two and a half years to complete and involved 20,344 users who had donated their computing power.

Mathematicians believe that there are infinitely many such twin primes: that's the famous twin prime conjecture. It has been around for over 150 years but so far no one has been able to prove that it is true. What is worse, until a few months ago, it was not even proved that there is any finite number N so that there are infinitely many primes that are N apart (the twin prime conjecture would correspond to N=2).

Back in April Yitang Zhang, a little known mathematician who had not published any research for over ten years, provided the first breakthrough: he showed that there is such a number N and that it is less than 70 million. "If this is right, I'm absolutely astounded," the mathematician Dorian Goldfeld told the journal Nature at the time. And after scrutiny by fellow mathematicians Zhang's proof was deemed to be correct.

The result caused a flurry of activity. It is hailed a landmark theorem not only because 70 million is a lot less than infinity, but also because it was hoped that the mathematical tools developed by Zhang could be used to push the size of the gaps down even further. When a new bound of under 5 million was announced in early June, Terrence Tao, whose work on prime numbers helped to earn him the acclaimed Fields medal in 2006, proposed a collaborative effort between mathematicians to try and go further. And the project seems to be a success with participants claiming they have reduced the bound all the way down to 4,680. It remains to be seen whether they can eventually reach the holy grail of 2.

Why should anyone care about a result like this? The standard answer to those asking for applications is that prime numbers have become essential in cryptography. They act as padlocks in the codes that encrypt your passwords or bank details whenever you send them over the internet (you can find out more in the Plus article Safety in numbers). The role of prime numbers in modern ecommerce could not have been foreseen and similarly it's hard to predict how and when current results about primes will come in useful.

But many mathematicians would quote another reason. What could be more natural than to explore the counting numbers 1, 2, 3, and so on? The prime numbers are the atoms among these numbers, the indivisible units, because every other positive whole number can be written as a product of primes in a unique way. This is why a lot of number theory boils down to understanding the primes. And as the mathematician Carl Friedrich Gauss put it, "Mathematics is the queen of the sciences and number theory is the queen of mathematics".