Number Theory
On the sum of distinct primes or squares of primes
Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 647-649.

In 1965 Erdős introduced $f2(s)$: $f2(s)$ is the smallest integer such that every $l>f2(s)$ 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, $f2(s)⩽p2+p3+⋯+ps+1+3106$, and the set of s with the equality has the density 1.

En 1965 Paul Erdős a introduit la valeur $f2(s)$ comme le plus petit entier tel que tout entier $l>f2(s)$ 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 $f2(s)⩽p2+p3+⋯+ps+1+3106$ et que lʼensemble des s réalisant lʼégalité est de densité 1.

Accepted:
Published online:
DOI: 10.1016/j.crma.2012.08.003

Jin-Hui Fang 1; Yong-Gao Chen 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
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Jin-Hui Fang; Yong-Gao Chen. On the sum of distinct primes or squares of primes. Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 647-649. doi : 10.1016/j.crma.2012.08.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.08.003/

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

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

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

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

Cited by 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.