Répartition modulo 1 de la suite
For an integer , let be the generalized Pell sequence which starts with ( terms) and each term afterwards is given by the linear recurrence . In this paper, we find all -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence .
The sequence of balancing numbers is defined by the recurrence relation for with initial conditions and is called the th balancing number. In this paper, we find all repdigits in the base which are sums of four balancing numbers. As a result of our theorem,...
désigne la somme des chiffres de l’entier en base et la somme des chiffres de associée au développement de en fraction continue. Dans un article paru aux Annales de l’Institut Fourier (31 (1981), 1–15), Coquet, Rhin et Toffin montrent que, lorsque ou est irrationnel, la suite est équirépartie modulo 1. On précise ici que l’équirépartition est uniforme.
A representation field for a non-maximal order in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of . Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.
Let be the set of integers, the set of nonnegative integers and be a binary linear form whose coefficients , are nonzero, relatively prime integers such that and . Let be any function such that the set has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set of integers such that for all integers , where . We add the structure of difference for the binary linear form .
For any given positive integer k, and any set A of nonnegative integers, let denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both and hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.