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
This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, it would be true. (Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7).
In 2013, Harald Helfgott proved Goldbach's weak conjecture; previous results had already shown it to be true for all odd numbers greater than
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. Helfgott's proof covers both versions of the conjecture.
Timeline of results
In 1923, Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the odd 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. Borozdin proved that 3315 is large enough. This number has 6,846,169 decimal digits, so checking every number under this figure would be completely infeasible.
In 1997, Deshouillers, Effinger, te Riele and Zinoviev published a result showing 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.
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. In 2012, Terence Tao proved this without the Riemann Hypothesis; this improves both results.
In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered this threshold to approximately
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. Here, the major arcs