Goldbach's conjecture

Statement

The conjecture has the following equivalent forms:

• Every even integer greater than $2$ is expressible as a sum of two (possibly equal) primes.
• Every even integer greater than $4$ 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

• Goldbach's weak conjecture is the statement that every odd integer greater than $7$ is expressible as the sum of three odd primes. This is currently known to follow from the generalized Riemann hypothesis.
  • Vinogradov's theorem states that every sufficiently large odd integer is expressible as the sum of three odd primes. The current bound of sufficiently large is approximately $e^{3100}\simeq 2\cdot 10^{1346}$.
  • Chaohua's strengthening of Vinogradov's theorem states that we can choose these odd primes to be roughly equal.
• Schnirelmann's theorem on Goldbach's conjecture: This states that every even integer greater than $2$ is expressible as the sum of at most $300,000$ primes.
• Chen's theorem which states that 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.