Goldbach's weak conjecture

Goldbach's weak conjecture

Letter from Goldbach to Euler dated on 7 June 1742 (Latin-German)[1]
Field	Number theory
Conjectured by	Christian Goldbach
Conjectured in	1742
First proof by	Harald Helfgott
First proof in	2013
Implied by	Goldbach's conjecture

In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, is the proposition that every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum.)

This conjecture is called "weak" because it has been shown that a proof for Goldbach's strong conjecture (concerning sums of two primes) would have this weak conjecture as a corollary, since, if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3).

In 2013, Harald Helfgott released a proof of Goldbach's weak conjecture.^[2] The proof was accepted for publication in the Annals of Mathematics Studies series^[3] in 2015, and has been undergoing further review and revision since; fully refereed chapters in close to final form are being made public in the process.^[4] If and when the proof is accepted, it will promote the conjecture to the status of theorem.

Some state the conjecture as

Every odd number greater than 7 can be expressed as the sum of three odd primes.^[5]

This version excludes 7 = 2+2+3, as 7 requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+13, which are allowed in the other formulation. Helfgott's proof covers both versions of the conjecture. Like the other formulation, this one also immediately follows from Goldbach's strong conjecture.