
Glossary: Prime Pages: Top 5000: 
GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) The odd Goldbach conjecture (sometimes called the 3primes problem) is that "every odd integer greater than five is the sum of three primes." Compare this with Goldbach's conjecture: every even integer greater than two is the sum of two primes. If the Goldbach's conjecture is true, then so is the odd Goldbach conjecture. There has been substantial progress on the odd Goldbach conjecture, the easier case of Goldbach's conjecture. In 1923 Hardy and Littlewood [HL23] showed that it follows from the Riemann Hypothesis for all sufficiently large integers. In 1937 Vinogradov [Vinogradov37] removed the dependence on the Riemann Hypothesis, and proved that this it true for all sufficiently large odd integers n (but did not quantify "sufficiently large"). In 1956 Borodzkin showed n greater than 3^{14348907} is sufficient in Vinogradov's proof. In 1989 Chen and Wang reduced this bound to 10^{43000}; and later to 10^{7194} [CW1996]. The exponent still must be reduced dramatically before we can use computers to take care of all the smaller cases. Zinoviev showed that if we are willing to accept the Generalized Riemann Hypothesis (GRH), then this exponent can be reduced to just 10^{20}. Using an estimate by Schoenfeld; a paper by Deshouillers, Effinger, Te Riele and Zinoviev (1997) showed that it is enough (given the GRH) to check the even integers less than 1.615*10^{12} against Goldbach's (two prime) conjecture, which they did! So then, once the Generalized Riemann Hypothesis is proved, the odd Goldbach conjecture will be too.
See Also: GoldbachConjecture References:
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