# Goldbach's conjecture

From Number

Template:Additive partition conjecture

## Statement

The conjecture has the following equivalent forms:

- Every even integer greater than is expressible as a sum of two (possibly equal) primes.
- Every even integer greater than is expressible as a sum of two (possibly equal) odd primes.

A partition of an even integer as a sum of two primes is termed a *Goldbach partition*.

## Relation with other facts and conjectures

### Weaker conjectures and facts

Name of conjecture/fact | Statement | Status |
---|---|---|

Weak Goldbach conjecture | every odd integer greater than is expressible as the sum of three odd primes | corollary of generalized Riemann hypothesis |

Vinogradov's theorem | every sufficiently large odd integer is expressible as the sum of three odd primes. Also finds that there are many such triples. The current bound of sufficiently large is approximately |
proved |

Haselgrove's strengthening of Vinogradov's theorem | in the statement of Vinogradov's theorem, we can choose the three primes to be roughly equal |
proved |

Chaohua's strengthening of Vinogradov's theorem | numerically strengthens Haselgrove's statement | proved |

Schnirelmann's theorem on Goldbach's conjecture | every even integer greater than is expressible as the sum of at most primes | proved |

Chen's theorem on Goldbach's conjecture | every sufficiently large even integer is expressible as the sum of a prime number and a semiprime, i.e., a number that is either prime or is a product of two primes | proved |