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Let $α > 1$ be irrational. Several authors studied the numbers $ℓ^{m} (α) = inf {| y | : y \in Λ_{m}, y \neq 0},$ where $m$ is a positive integer and $Λ_{m}$ denotes the set of all real numbers of the form $y = ϵ_{0} α^{n} + ϵ_{1} α^{n - 1} + \dots + ϵ_{n - 1} α + ϵ_{n}$ with restricted integer coefficients $| ϵ_{i} | \leq m$ . The value of $ℓ^{1} (α)$ was determined for many particular Pisot numbers and $ℓ^{m} (α)$ for the golden number. In this paper the value of $ℓ^{m} (α)$ is determined for irrational numbers $α$ , satisfying $α^{2} = a α \pm 1$ with a positive integer $a$ .

An arithmetic formula of Liouville

Erin McAfee, Kenneth S. Williams (2006)

Journal de Théorie des Nombres de Bordeaux

An elementary proof is given of an arithmetic formula, which was stated but not proved by Liouville. An application of this formula yields a formula for the number of representations of a positive integer as the sum of twelve triangular numbers.

An arithmetic function

K. Nageswara Rao (1971)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

An arithmetical mapping and applications to Ω-results for the Riemann zeta function

Titus Hilberdink (2009)

Acta Arithmetica

An Artin Character and Representations of Primes by Binary Quadratic Forms III.

Halter-Koch, Franz (1985)

Manuscripta mathematica

An awful problem about integers in base four

J. Loxton, Alfred van der Poorten (1987)

Acta Arithmetica

An effective procedure for minimal bases of ideals in Z[x]

Luis F. Cáceres-Duque (2003)

Discussiones Mathematicae - General Algebra and Applications

We give an effective procedure to find minimal bases for ideals of the ring of polynomials over the integers.

An Egyptian algorithm for polynomials.

D.E. Dobbs, R.M. McConnel (1984)

Elemente der Mathematik

An elementary proof of a congruence by Skula and Granville

Romeo Meštrović (2012)

Archivum Mathematicum

Let $p \geq 5$ be a prime, and let $q_{p} (2) : = (2^{p - 1} - 1) / p$ be the Fermat quotient of $p$ to base $2$ . The following curious congruence was conjectured by L. Skula and proved by A. Granville $q_{p} {(2)}^{2} \equiv - \sum_{k = 1}^{p - 1} \frac{2^{k}}{k^{2}} (mod p) .$ In this note we establish the above congruence by entirely elementary number theory arguments.

An exercise on Fibonacci representations

Jean Berstel (2001)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We give a partial answer to a question of Carlitz asking for a closed formula for the number of distinct representations of an integer in the Fibonacci base.

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An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers.

An analysis of GCD and LCM matrices via the $L D L^{T}$ -factorization.

An application of a formula of Western to the evaluation of certain Jacobsthal sums

An application of dihedral fields to representations of primes by binary quadratic forms

An application of Kloosterman sums

An approximation property of quadratic irrationals

An arithmetic formula of Liouville

An arithmetic function

An arithmetical mapping and applications to Ω-results for the Riemann zeta function

An Artin Character and Representations of Primes by Binary Quadratic Forms III.

An awful problem about integers in base four

An effective procedure for minimal bases of ideals in Z[x]

An Egyptian algorithm for polynomials.

An elementary proof of a congruence by Skula and Granville

An exercise on Fibonacci representations

An extension of a theorem of Nielsen

An extension of Daykin's generalized Möbius function to unitary divisors.

An identity for a class of arithmetical functions of several variables.

An identity for a class of arithmetical functions of two variables.

An identity involving multiplicative orders.

Currently displaying 161 – 180 of 1815