SOME DIVISIBILITY PROPERTIES OF THE EULER FUNCTION

William D. Banks, Florian Luca, Igor E. Shparlinski

Type: Article

Publication Date: 2005-11-29

Citations: 2

DOI: https://doi.org/10.1017/s0017089505002752

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Let $\varphi(\cdot)$ denote the Euler function, and let $a>1$ be a fixed integer. We study several divisibility conditions which exhibit typographical similarity with the standard formulation of the Euler theorem, such as $a^n \equiv 1\!\!\!\!\pmod{\varphi(n)}$, and we estimate the number of positive integers $n\le x$ satisfying these conditions.

Locations

Glasgow Mathematical Journal - View - PDF
MOspace Institutional Repository (University of Missouri) - View - PDF

Similar Works

Action	Title	Year	Authors
+	A characterization of Euler’s constant	2013	Horst Alzer
+	A formula of Euler	1993	T. W. Körner
+	A solution to an open problem on the Euler–Mascheroni constant	2013	Ioan Gavrea Mircea Ivan
+	A generalization of the Eulerian numbers	2022	Lajos Takács
+	An analogue of Euler’s ϕ-function	1944	E. K. Haviland
+	A Generalization of Euler’s ɸ-Function	1983	Paula Fernández Steve Ligh
+	A generalization of Euler’s constant	2007	Alina Sı̂ntămărian
+	Arithmetical properties of the Euler’s function and generalizations	1998	Frédéric Marie-Jeanne
+	Euler’s Formula and Applications	2015
+	Euler’s Formula and Applications	2018
+	A Development of Euler Numbers	1997	Joseph B. Dence
+ PDF Chat	A Divisibility Property of the Bernoulli Polynomials	1952	L. Carlitz
+ PDF Chat	A divisibility property of the Bernoulli polynomials	1952	L. Carlitz
+	On the Euler-Mascheroni constant	2013	Juozas Juvencijus Mačys
+	A New Inequality for Eulerian Numbers	2012	Yilun Shang
+	A note on Euler's p-function	2011	Władysław Narkiewicz
+	An equation involving the Euler function	2007	Guohui Chen
+	A Nest of Euler inequalities	2011	Qi Luo
+	A Note on Euler’s Numbers	1991	Nenad Cakić
+	Chapter 4 Euler's Formula and its Consequences	1967

Works That Cite This (2)

Action	Title	Year	Authors
+	On the proportion of numbers coprime to a given integer	2008	Paul Erdős Florian Luca Carl Pomerance
+ PDF Chat	Arithmetic properties of coefficients of L-functions of elliptic curves	2018	Ahmet M. Güloğlu Florian Luca Aynur Yalçıner

Works Cited by This (14)

Action	Title	Year	Authors
+ PDF Chat	Multiplicative structure of values of the Euler function	2004	William D. Banks John Friedlander Carl Pomerance Igor E. Shparlinski
+	On the Normal Behavior of the Iterates Of some Arithmetic Functions	1990	Paul Erdős Andew Granvilie Carl Pomerance Claudia Spiro
+	Weylsche Exponentialsummen in der neueren Zahlentheorie	1963	Arnold Walfisz
+	Introduction to Analytic and Probabilistic Number Theory	2015	Gérald Tenenbaum
+ PDF Chat	Shifted primes without large prime factors	1998	Roger C. Baker G. Harman
+	On the number of positive integers ≦ x and free of prime factors > y	1986	Adolf Hildebrand
+	A New Proof of a Theorem of Van Der Corput	1989	Bernard Landreau
+ PDF Chat	On two congruences for primality	1974	M. V. Subba Rao
+ PDF Chat	On composite n for which φ(n) \| n -1	1976	Carl Pomerance
+ PDF Chat	Integers without large prime factors	1993	Adolf Hildebrand Gérald Tenenbaum