Goldbach's Weak Conjecture

Conjecture
Every odd integer greater than $$7$$ is the sum of three odd primes.

It is also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem

Progress

 * 1923: Proved by Hardy and Littlewood that the Generalized Riemann Hypothesis implies Goldbach's Weak Conjecture for sufficiently large numbers.


 * 1937: Proved by Vinogradov, independently of the Generalized Riemann Hypothesis that all sufficiently large numbers can be expressed as the sum of three primes.


 * 1939: Vinogradov's student K. Borozdin proved that $$3^{14348907}$$ is large enough.


 * 1997: Proved by Deshouillers, Effinger, te Riele and Zinoviev that the Generalized Riemann Hypothesis implies Goldbach's Weak Conjecture.


 * 2002: Liu Ming-Chit and Wang Tian-Ze lowered the threshold for Vinogradov's result to approximately $$n > e^{3100}$$, which is roughly $$2 \times 10^{1346}$$. This limits the threshold so that it is feasible to test any single odd number below that threshold, so to a certain extent it is nowonly a matter of time.

Also see

 * Goldbach Conjecture


 * Goldbach Conjecture implies Weak Goldbach Conjecture, which is why this one is called weak.