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Prime numbers with Beatty sequences

William D. Banks, Igor E. Shparlinski (2009)

Colloquium Mathematicae

A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form p = 2⌊αn⌋ + 1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic...

Primitive Points on a Modular Hyperbola

Igor E. Shparlinski (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U] × [1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v) =1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are "visible" from the origin.

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

We consider Weil sums of binomials of the form W F , d ( a ) = ∑ x ∈ F ψ ( x d - a x ) , where F is a finite field, ψ: F → ℂ is the canonical additive character, g c d ( d , | F × | ) = 1 , and a ∈ F × . If we fix F and d, and examine the values of W F , d ( a ) as a runs through F × , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo | F × | ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n odd, and d = 3 r + 2 with...

Propriétés arithmétiques des substitutions et automates infinis

Christian Mauduit (2006)

Annales de l’institut Fourier

L’objet de ce travail est d’étudier les propriétés arithmétiques et statistiques des mots infinis et des suites de nombres entiers engendrés par des substitutions sur un alphabet infini ou par des automates déterministes ayant un nombre infini dénombrable d’états. En particulier, nous montrons que si u est une suite de nombres entiers engendrée par un automate dont le graphe étiqueté associé représente une marche aléatoire de moyenne nulle sur un réseau de ℝ d ( d entier positif), alors la suite ( n α ) n ∈ u ...

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