# 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 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:**

- CW1996
Chen, Jing RunandWang, Tian Ze, "The Goldbach problem for odd numbers,"Acta Math. Sinica (Chin. Ser.),39:2 (1996) 169--174.MR1411958(Abstract available)- CW89
J. R. ChenandY. Wang, "On the odd Goldbach problem,"Acta Math. Sinica,32(1989) 702--718.- DERZ97
J. M. Deshouillers,G. Effinger,H. te RieleandD. 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 RieleandY. 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. HardyandJ. 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 10^{20},"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 integersn]