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. "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, France: Secrétariat Math., pp. 1-6, 1977.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, France: Secrétariat Mathématique,
pp. 1-4, 1973.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.
