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SOME DIVISIBILITY PROPERTIES OF THE EULER FUNCTION

Published online by Cambridge University Press:  29 November 2005

WILLIAM D. BANKS ,
FLORIAN LUCA  and
IGOR E. SHPARLINSKI
WILLIAM D. BANKS
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211 USA e-mail: [email protected]
FLORIAN LUCA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, México e-mail: [email protected]
IGOR E. SHPARLINSKI
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia e-mail: [email protected]
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Abstract

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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.

Keywords

11A0711N37
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 47 , Issue 3 , September 2005 , pp. 517 - 528
Copyright
2005 Glasgow Mathematical Journal Trust