Goldbach's conjecture

Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:

Every even integer greater than 2 can be written as the sum of two primes.

Origins

On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he suggested the following conjecture, which would later be called Goldbach's strong conjecture:

Every integer greater than 2 can be written as the sum of two primes.

He considered 1 to be a prime number, a convention subsequently abandoned. Goldbach wrote that even numbers 4 and up could always be composed of two different prime numbers.

A weaker version of Goldbach's original conjecture is:

Every integer greater than 5 can be written as the sum of three primes.

This is called Goldbach's weak conjecture. Euler, becoming interested in the problem, wrote back to Goldbach saying that the weak conjecture would be implied by Goldbach's strong conjecture, saying that he was certain that the theorem was true ("ein ganz gewisses Theorema"), but he was unable to prove it.

Goldbach's weak conjecture was later proved by Harald Helfgott in 2013, but Goldbach's strong conjecture has not been proved yet.

Other websites

• Goldbach's conjecture, part of Chris Caldwell's Prime Pages.