odd Goldbach conjecture

The odd Goldbach conjecture (sometimes called the 3-primes 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 10⁴³⁰⁰⁰; and later to 10⁷¹⁹⁴ [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 10²⁰. 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*10¹² 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:

CW1996

Chen, Jing Run and Wang, Tian Ze, "The Goldbach problem for odd numbers," Acta Math. Sinica (Chin. Ser.), 39:2 (1996) 169--174.  MR1411958 (Abstract available)

CW89

J. R. Chen and Y. Wang, "On the odd Goldbach problem," Acta Math. Sinica, 32 (1989) 702--718.

DERZ97

J. M. Deshouillers, G. Effinger, H. te Riele and D. Zinoviev, "A complete Vinogradov 3-primes theorem under the Riemann hypothesis," ERA Amer. Math. Soc., 3 (1997) 94--104.  MR 98g:11112 (Abstract available)

DRS98

J. M. Deshouillers, H. J. J. te Riele and Y. Saouter, New experimental results concerning the Goldbach conjecture.  In "Proc. 3rd Int. Symp. on Algorithmic Number Theory," Lecture Notes in Computer Science Vol, 1423, 1998.  pp. 204--215, MR 2000j:11143

HL23

G. H. Hardy and J. E. Littlewood, "Some problems of `partitio numerorum' : III: on the expression of a number as a sum of primes," Acta Math., 44 (1923) 1-70.  Reprinted in "Collected Papers of G. H. Hardy," Vol. I, pp. 561-630, Clarendon Press, Oxford, 1966.

Saouter98

Y. Saouter, "Checking the odd Goldbach conjecture up to 10²⁰," Math. Comp., 67 (1998) 863-866.  MR 98g:11115 (Abstract available)

Vinogradov37

I. M. Vinogradov, "Representation of an odd number as the sum of three primes," Dokl. Akad. Nauk SSSR, 16 (1937) 179--195.  Russian. [Proves that the odd Goldbach conjecture holds for sufficiently all large integers n]