On the sum of distinct primes or squares of primes

Fang, Jin-Hui; Chen, Yong-Gao

doi:10.1016/j.crma.2012.08.003

Number Theory

On the sum of distinct primes or squares of primes
[Sur les sommes de premiers et de carrés de premiers distincts]

Présenté par : the Editorial Board

Fang, Jin-Hui ¹ ; Chen, Yong-Gao ²

¹ Department of Mathematics, Nanjing University of Information Science & Technology, Nanjing 210044, PR China
² School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, PR China

Comptes Rendus. Mathématique, Tome 350 (2012) no. 13-14, pp. 647-649.

Résumé
Abstract

En 1965 Paul Erdős a introduit la valeur $f_{2} (s)$ comme le plus petit entier tel que tout entier $l > f_{2} (s)$ est la somme de s premiers ou carrés de premiers distincts, où un nombre premier et son carré ne sont simultanément utilisés. Nous démontrons que pour tout s suffisamment grand on a $f_{2} (s) ⩽ p_{2} + p_{3} + \dots + p_{s + 1} + 3106$ et que lʼensemble des s réalisant lʼégalité est de densité 1.

In 1965 Erdős introduced $f_{2} (s)$ : $f_{2} (s)$ is the smallest integer such that every $l > f_{2} (s)$ is the sum of s distinct primes or squares of primes where a prime and its square are not both used. We prove that for all sufficiently large s, $f_{2} (s) ⩽ p_{2} + p_{3} + \dots + p_{s + 1} + 3106$ , and the set of s with the equality has the density 1.

Reçu le : 2012-05-06
Accepté le : 2012-08-10
Publié le : 2012-07-01

DOI : 10.1016/j.crma.2012.08.003

Affiliations des auteurs :

Fang, Jin-Hui ¹ ; Chen, Yong-Gao ²

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Fang, Jin-Hui; Chen, Yong-Gao. On the sum of distinct primes or squares of primes. Comptes Rendus. Mathématique, Tome 350 (2012) no. 13-14, pp. 647-649. doi : 10.1016/j.crma.2012.08.003. http://www.numdam.org/articles/10.1016/j.crma.2012.08.003/

Bibliographie
Cité par

[1] V. Brun, Le crible dʼEratosthene et le théorème de Goldbach, Videnskapselkapets Skrifter, I, No. 3, Kristiania, 1920.

[2] Chen, Y.-G. The analogue of Erdős–Turán conjecture in $Z_{m}$ , J. Number Theory, Volume 128 (2008), pp. 2573-2581

[3] Erdős, P. On a problem of Sierpiński, Acta Arith., Volume 11 (1965), pp. 189-192

[4] Sierpiński, W. Sur les suites dʼentiers deux á deux premiers entre eux, Enseign. Math., Volume 10 (1964), pp. 229-235

Cité par Sources :

^☆ This work was supported by the National Natural Science Foundation of China, Grant Nos. 11071121, 11201237 and the Youth Foundation of Mathematical Tianyuan of China, Grant No. 11126302. The first author is also supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions, Grant No. 11KJB110006.