The constant
in Schnirelmann's theorem such that every
integer is a sum of at most primes. Of course, by Vinogradov's theorem, it is known that 4 primes
suffice for all sufficiently large numbers, but this constant gives a sufficient
number for all numbers. The best current estimate is (Ramaré 1995), and a summary of progress on upper
bounds for
is summarized in the following table.
Deshouillers, J.-M. No. 17 in "Amélioration de la constante de Šnirelman dans le probléme de Goldbach." Séminaire
Delange-Pisot-Poitou (14e année: 1972/73). Théorie des nombres: Fascicule
2: Exposés 17 à 26, et Groupe d'étude. Paris: Secrétariat
Mathématique, pp. 1-4, 1973.Deshouillers, J.-M. "Sur
la constante de Šnirel'man." No. G16 in Séminaire Delange-Pisot-Poitou,
17e année (1975/76). Théorie des nombres: Fascicule 2: Exposés
23 à 31 et Groupe d'étude. Paris: Secrétariat Math., pp. 1-6,
1977.Klimov, K. I. Naucn. Trudy Kuibysev Gos. Ped. Inst.158,
14-30, 1975.Klimov, N. I.; Pil'tjaĭ, G. Z.; and Šeptickaja,
T. A. "An Estimate of the Absolute Constant in the Goldbach-Šnirel'man
Problem." In Issledovaniya po teorii chisel, Vyp. 4. [Studies
in number theory, No. 4] (Ed. N. Lenskoĭ). Saratov: Izdat. Saratov.
Univ., pp. 35-51, 1972.Ramaré, O. "On Šnirel'man's
Constant." Ann. Scuola Norm. Sup. Pisa Cl. Sci.22, 645-706, 1995.Riesel,
H. and Vaughan, R. C. "On Sums of Primes." Ark. Mat.21,
46-74, 1983.Vaughan, R. C. "On the Estimation of Schnirelman's
Constant." J. reine angew. Math.290, 93-108, 1977.
in Schnirelmann's theorem such that every
integer 
is a sum of at most 
primes. Of course, by Vinogradov's theorem, it is known that 4 primes
suffice for all sufficiently large numbers, but this constant gives a sufficient
number for all numbers. The best current estimate is 
(Ramaré 1995), and a summary of progress on upper
bounds for 
is summarized in the following table.
