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 gives an upper bound such that it is feasible to test any single odd number below that threshold. Though one can calculate: if a modern desktop were the size of a 1cm cube, one filled the solar system with such cubes and left them running the fastest known algorithms since the beginning of time, numbers larger than $10^{70}$ would remain untested. For this reason the problem is still considered unsolved, though the existence of only finitely many counterexamples is a worthwhile result in its own right.

Also see

 * Goldbach Conjecture
 * Goldbach's Marginal Conjecture


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