# Twin Prime Conjecture

### From Mathsreach

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*