# Goldbach Conjecture

## Conjecture

Every even integer greater than $2$ is the sum of two primes.

### Marginal Conjecture

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

## Hilbert $23$

This problem is no. $8b$ in the Hilbert $23$.

## Landau's Problems

This is the $1$st of Landau's problems.

## Source of Name

This entry was named for Christian Goldbach.

## Historical Note

Christian Goldbach actually proposed what is now known as Goldbach's Marginal Conjecture in $1742$ in a letter to Leonhard Paul Euler. Euler then proposed this stronger conjecture.

It was published in Edward Waring's Meditationes Algebraicae of $1770$.

It has been checked by computer for numbers up to at least $10^{18}$.

In $1937$ Ivan Matveevich Vinogradov proved Vinogradov's Theorem: that all sufficiently large odd numbers are the sum of $3$ primes.

In $1973$ Chen Jingrun proved Chen's Theorem: that every sufficiently large even number is the sum of a prime and either another prime or a semiprime.

In $1995$ Olivier Ramaré showed in Ramaré's Theorem that every even number is the sum of no more than $6$ primes.