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