Twin Prime Conjecture

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There are infinitely many primes p such that p + 2 is also prime.

Simply: That the number of prime numbers that differ by two, such as 101 and
103, is infinite.

Discipline: Number theory.

Originator: Euclid, around 300BC.

Incentive: Being the first to solve a 2,300- year-old problem.

Partial proofs: In 1915, Norwegian mathematician Viggo Brun showed that the
sum of reciprocals of the twin primes was convergent. This famous result was the first
use of the Brun sieve and helped initiate the development of modern sieve theory, a
set of techniques designed to estimate the size of sifted sets of integers.

From 1940, Paul Erdős showed that there is a constant c < 1 and infinitely many
primes p such that (p′ - p) < (c ln p) where p′ denotes the next prime after p. This
result was successively improved by Helmut Maier, Daniel Goldston and Cem Yildirim.

In 1966, Chinese mathematician Chen Jingrun used sieve theory to show that
there are infinitely many primes p such that p + 2 is either a prime or the product of
two primes, now known as Chen primes. Terence Tao and Ben Green built on this
to show that there are infinitely many three-term arithmetic progressions of
Chen primes.

Mathematicians believe the twin prime conjecture to be true, based on numerical
evidence and the probabilistic distribution of primes.

Unusual aspect: Because it is easily understood by non-mathematicians, the
twin prime conjecture is a popular target for pseudo-mathematicians who attempt to
prove or disprove it, sometimes using only high-school mathematics.

NZIMA connection: Marcus du Sautoy, visiting Maclaurin Fellow in 2007, for
whom the distribution of prime numbers is a major interest.

Published in IMAges 4 - April 2008