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A Note on Šnirel'man's Approach to Goldbach's Problem

A Note on Šnirel'man's Approach to Goldbach's Problem A NOTE ON SNIREL'MAN'S APPROACH TO GOLDBACH'S PROBLEM R. C. VAUGHAN 1. Introduction and statement of theorem The object of this note is to demonstrate that Snirel'man's method [38], [39] (see also Landau [22], [23]) as modified by Shapiro and Warga [36] will give the following theorem. THEOREM. Every sufficiently large odd number can be written as the sum of five odd prime numbers, and thus every sufficiently large number is the sum of at most six prime numbers. It is apparent from the proof that the method only just misses giving four for even numbers, and that is the best that has been obtained for even numbers by the much more powerful Hardy-Littlewood-Vinogradov method (see I. M. Vinogradov [42], [43], [44]). In [38] Snirel'man has a large positive constant in place of the six, and this has been reduced successively to 2208 by Romanov [35], to 71 by Heilbronn, Landau and Scherk [17], 67 by Ricci [32], [33], 20 by Shapiro and Warga [36], 18 by Yin [45], 12 by Klimov and Kondakova [20] and 10 by CeSuro and KuzjaSev [5] and Siebert [37] independently. CeCuro and KuzjaSev, and Siebert use the Bombieri-A. I. Vinogradov theorem [4], [41], http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Note on Šnirel'man's Approach to Goldbach's Problem

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/8.3.245
Publisher site
See Article on Publisher Site

Abstract

A NOTE ON SNIREL'MAN'S APPROACH TO GOLDBACH'S PROBLEM R. C. VAUGHAN 1. Introduction and statement of theorem The object of this note is to demonstrate that Snirel'man's method [38], [39] (see also Landau [22], [23]) as modified by Shapiro and Warga [36] will give the following theorem. THEOREM. Every sufficiently large odd number can be written as the sum of five odd prime numbers, and thus every sufficiently large number is the sum of at most six prime numbers. It is apparent from the proof that the method only just misses giving four for even numbers, and that is the best that has been obtained for even numbers by the much more powerful Hardy-Littlewood-Vinogradov method (see I. M. Vinogradov [42], [43], [44]). In [38] Snirel'man has a large positive constant in place of the six, and this has been reduced successively to 2208 by Romanov [35], to 71 by Heilbronn, Landau and Scherk [17], 67 by Ricci [32], [33], 20 by Shapiro and Warga [36], 18 by Yin [45], 12 by Klimov and Kondakova [20] and 10 by CeSuro and KuzjaSev [5] and Siebert [37] independently. CeCuro and KuzjaSev, and Siebert use the Bombieri-A. I. Vinogradov theorem [4], [41],

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1976

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