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Carmichael numbers composed of primes from a Beatty sequence

William D. Banks, Aaron M. Yeager (2011)

Colloquium Mathematicae

Let α,β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence ℬ α , β = ( ⌊ α n + β ⌋ ) n = 1 ∞ . We conjecture that the same result holds true when α is an irrational number of infinite type.

Chebotarev sets

Hershy Kisilevsky, Michael O. Rubinstein (2015)

Acta Arithmetica

We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give necessary conditions for a set to be realized in this manner, and show that the subset of primes consisting of every other prime cannot be expressed in this way, even if we allow a finite number of exceptions.

Computing higher rank primitive root densities

P. Moree, P. Stevenhagen (2014)

Acta Arithmetica

We extend the "character sum method" for the computation of densities in Artin primitive root problems given by Lenstra and the authors to the situation of radical extensions of arbitrary rank. Our algebraic set-up identifies the key parameters of the situation at hand, and obviates the lengthy analytic multiplicative number theory arguments that used to go into the computation of actual densities. It yields a conceptual interpretation of the formulas obtained, and enables us to extend their range...

Concentration function of additive functions on shifted twin primes

Simon Wong (1998)

Acta Arithmetica

0. Introduction. The content of this paper is part of the author's Ph.D. thesis. The two new theorems in this paper provide upper bounds on the concentration function of additive functions evaluated on shifted γ-twin prime, where γ is any positive even integers. Both results are generalizations of theorems due to I. Z. Ruzsa, N. M. Timofeev, and P. D. T. A. Elliott.

Écarts entre nombres premiers successifs

Emmanuel Kowalski (2005/2006)

Séminaire Bourbaki

Le théorème des nombres premiers dit que la distance entre deux nombres premiers consécutifs p n < p n + 1 est, en moyenne, de l’ordre de ln ( p n ) . Récemment, D. Goldston, J. Pintz et C. Yıldırım sont parvenus à démontrer que la distance normalisée ( p n + 1 - p n ) / ln ( p n ) pouvait devenir arbitrairement petite, améliorant spectaculairement les résultats connus auparavant. Sous des hypothèses considérées comme raisonnables, ils parviennent à montrer que p n + 1 - p n < 16 infiniment souvent. Leur méthode est une très jolie application d’idées inspirée par...

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