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, states 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.)
Letter from Goldbach to Euler dated on 7 June 1742 (Latin-German) | |
| Field | Number theory |
|---|---|
| Conjectured by | Christian Goldbach |
| Conjectured in | 1742 |
| First proof by | Harald Helfgott |
| First proof in | 2013 |
| Implied by | Goldbach's conjecture |
This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, then this would also be true. For 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. As of 2018, the proof is widely accepted in the mathematics community, but it has not yet been published in a peer-reviewed journal. 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.
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 because this 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.