Also known as Goldbach's Weak conjecture, and related to Goldbach's Conjecture, this problem asks the following: Is every odd number greater than five, the sum of three primes? This problem, while similar to Goldbach's Conjecture, has proven somewhat easier to attack, and very substantial progress has been made. In 1923 Hardy and Littlewood showed that it follows from the Riemann Hypothesis for all sufficiently large integers (though there was no bound proven.) In 1937 Vinogradov removed the dependence on the Riemann Hypothesis, and proved that this it true for all sufficiently large odd integers n. In 1956 Borodzkin showed n greater than 314348907 is sufficient in Vinogradov's proof. In 1989 Chen and Wang reduced this bound to 1043000.

Zinoviev showed that if we are willing to accept the Generalized Riemann Hypothesis, then this exponent can be reduced to just 1020. Using an estimate by Schoenfeld; a paper by Deshouillers, Effinger, Te Riele and Zinoviev showed that it is enough (given the Generalized Riemann Hypothesis) to check the even integers less than 1.615*1012 against Goldbach's conjecture, which they did. If the Generalized Riemann hypothesis happens to be proven before someone can test all of the numbers until 1043000, or the bound is lowered, it will be proven.

Wolfram's World of Mathematics,

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