Are there Infinitely Many Twin Primes?

Recently, there has been much progress towards proving the twin prime conjecture, one of the great open questions in number theory. Let’s go on a little trip towards the latest developments together.

Primes

A prime number is a positive integer greater than \(1\) that has no positive divisors other than \(1\) and itself.

There are infinitely many primes, a fact that was already known by the ancient Greek. A short proof, although not the classical one by Euclid from 300 BC, is the following. Assume that there are only a finite number of primes. This implies that there is a largest prime p. Consider the number \(p!+1\). This number is not divisible by any integer from \(2\) to \(p\) (each remainder is \(1\)). Hence, either \(p!+1\) is prime itself or it is divisible by a prime larger than \(p\), which is a contradiction since we had assumed that \(p\) was the largest prime. \(\square\)

The largest known prime at the time of writing is \(2^{57885161}-1\) (more than 17 million digits, discovered in January 2013 by GIMPS).

Twin Primes

Twin primes are pairs of primes of the form \((p,p+2)\). Examples are \((17,19)\) and \((197,199)\).

The largest known twin prime pair at the time of writing is \(3756801695685\times 2^{666669}\pm 1\) (more than 200000 digits, discovered in December 2011 by PrimeGrid).

A straightforward question now pops up: since there are an infinite number of primes, are there also an infinite number of twin primes? One of the reasons that this question is interesting, is that we know that the gap between the primes increases for larger numbers (from the prime number theorem, we learn that the “average” gap between primes smaller than \(p\) is \(\ln(p)\)). Nevertheless, is there an infinite number of twin primes anyway?

The Big Breakthrough

A big breakthrough towards solving the twin prime conjecture was made in 2013, when Yitang Zhang proved that there are an infinite number of prime pairs that are at most 70 million apart. In other words, he proved that there are an infinite number of pairs of primes of the form \((p,p+H\)), with \(H=70000000\). To a non-mathematician, this might not seem too impressive at first sight. However, it brings the gap down from unbounded to a fixed number, which is probably by far the hardest part.

The proof was quickly confirmed to be correct, and a flurry of activity followed to try to improve the result (i.e., prove the result for smaller values of \(H\)). A cool aspect of this cooperation is that it was done through a PolyMath project, and with remarkable success, since the limit was brought down to \(H=5414\) within two months. Have a look at the polymath blog for more information on these “massively collaborative online mathematical projects”.

Currently, the limit is down to \(H=246\). I look forward to it going all the way down to \(H=2\), although it is assumed that this is not within the reach of the current methods (although \(H=6\) might be attainable)…

[update] As of January 2016, the largest known prime is \(2^{74207281}-1\) (more than 22 million digits, also discovered by GIMPS).

[update 2] As of December 2017, the largest known prime is \(2^{77232917}-1\) (more than 23 million digits, also discovered by GIMPS).

[update 3] As of December 2018, the largest known prime is \(2^{82589933}-1\) (more than 24 million digits, also discovered by GIMPS).

[update 4] As of October 2024, the largest known prime is \(2^{136279841}-1\) (more than 41 million digits, also discovered by GIMPS).