Twin primes are pairs of prime numbers that differ by two, the first of which are 3 and 5, 5 and 7, 11 and 13, and 17 and 19. The largest example known, as of February 2003, is a pair of 51,090-digit primes discovered by Yves Gallot and Daniel Papp, with the value 33218925 × 2¹⁶⁹⁶⁹⁰±1. Other than the first, all twin primes have the form {6n – 1, 6n + 1}; also, the integers n and n + 2 form twin primes if and only if 4[(n – 1 )! + 1] = –n (mod n(n + 2)).

 

The twin prime conjecture is that there are infinitely many such pairings. It was first proposed in 1846 by French mathematician Alphonse de Polignac and so is sometimes referred to as the Polignac conjecture. No one made much progress on the theory of twin primes until Norwegian mathematician Viggo Brun came along in 1919. He showed that the sum of the reciprocals of the twin primes, (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19)… gets ever closer to a fixed number as bigger and bigger pairs are included. Brun's constant, as this sum became known, was calculated in 1976, using twin primes up to 100 billion, to be approximately 1.90216054. In 1994, American mathematician Thomas Nicely was trying to refine the value further using a personal computer based on Intel's then-new Pentium chip, when he noticed he was getting some strange results. The problem was traced to a flaw in the chip, which Intel subsequently fixed. So, interestingly, it was an attempt to solve a maths problem that led to the discvovery of a problem with the chip. In 2010 Nicely extended the known accuracy of Brun's constant to 1.902160583209 ± 0.000000000781 based on all twin primes less than 20 thousand trillion (2 × 10¹⁶).

 

An important step toward a proof of the twin primes conjecture came in 2003. American mathematician Daniel Goldston and Turkish mathematician Cem Yildirim showed that, if certain assumptions are made, there must be infinitely many twin primes that differ by no more than 16. Central among these assumptions is what's known as the Elliott-Halberstam conjecture – itself an open problem to do with the distribution of primes in arithmetic progressions (in which consecutive terms in a sequence of numbers differ by a constant amount). A mistake in their proof was corrected in 2005 with the help of Hungarian mathematician János Pintz. In 2013 American mathematician Yitang Zhan proved that, without making any assumptions at all, there's guaranteed to be an infinite number of prime pairs that differ by no more than 70 million. A year later this figure had been slashed to 246, and could be cut further to 12 or 6, respectively, if the Elliott-Halberstam conjecture or a generalised form of it were assumed to be correct.

 

The twin-prime conjecture generalizes to prime pairs that differ by any even number n, and generalizes even further to certain finite patterns of numbers separated by specified even differences. For example, the following triplets of primes all fit the pattern k, k + 2, and k + 6: 5, 7, and 11; 11, 13, and 17; 17, 19, and 23; 41, 43, and 47. It is believed that for any such pattern not outlawed by divisibility considerations there are infinitely many examples. (The pattern k, k + 2, and k + 4 has only one solution in primes, 3, 5, and 7, because any larger such triplet would contain a number divisible by 3.) Quartets of the form k, k + 2, k + 6, and k + 8 (the smallest example is 5, 7, 11, and 13) are thought to be infinite. For some patterns no example is known, or only one.