| Goldbach's weak conjecture | |
| Field: | Number theory |
| Conjectured By: | Christian Goldbach |
| Conjecture Date: | 1742 |
| First Proof By: | Harald Helfgott |
| First Proof Date: | 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, 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.)
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. The proof was accepted for publication in the Annals of Mathematics Studies series[1] 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.[2]
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.
See main article: Goldbach's conjecture.
The conjecture originated in correspondence between Christian Goldbach and Leonhard Euler. One formulation of the strong Goldbach conjecture, equivalent to the more common one in terms of sums of two primes, is
Every integer greater than 5 can be written as the sum of three primes.
The weak conjecture is simply this statement restricted to the case where the integer is odd (and possibly with the added requirement that the three primes in the sum be odd).
In 1923, Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the weak Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, Ivan Matveevich Vinogradov eliminated the dependency on the generalised Riemann hypothesis and proved directly (see Vinogradov's theorem) that all sufficiently large odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective Siegel–Walfisz theorem, did not give a bound for "sufficiently large"; his student K. Borozdkin (1956) derived that
| e16.038 | |
| e |
| 315 | |
| ≈ 3 |
In 1997, Deshouillers, Effinger, te Riele and Zinoviev published a result showing[4] that the generalized Riemann hypothesis implies Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 1020 with an extensive computer search of the small cases. Saouter also conducted a computer search covering the same cases at approximately the same time.[5]
Olivier Ramaré in 1995 showed that every even number n ≥ 4 is in fact the sum of at most six primes, from which it follows that every odd number n ≥ 5 is the sum of at most seven primes. Leszek Kaniecki showed every odd integer is a sum of at most five primes, under the Riemann Hypothesis.[6] In 2012, Terence Tao proved this without the Riemann Hypothesis; this improves both results.[7]
In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered Borozdkin's threshold to approximately
n>e3100 ≈ 2 x 101346
In 2012 and 2013, Peruvian mathematician Harald Helfgott released a pair of papers improving major and minor arc estimates sufficiently to unconditionally prove the weak Goldbach conjecture.[8] [9] [10] [11] [12] Here, the major arcs
akM
\left(a/q-cr0/qx,a/q+cr0/qx\right)
a/q,q<r0
c
ak{m}
ak{m}=(R/Z)\setminusak{M}