Goldbach's weak 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 (not necessarily distinct) primes.

This conjecture is called "weak" because it is implied by Goldbach's strong conjecture concerning sums of two primes: indeed, 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 supposed proof of Goldbach's weak conjecture. The proof was accepted for publication in the Annals of Mathematics Studies series 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. If 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.

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.