# 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.

## Also see

## 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.

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Goldbach's conjecture** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Goldbach's conjecture** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers: Fermat