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1995 Exceptional units and numbers of small Mahler measure
Joseph H. Silverman
Experiment. Math. 4(1): 69-83 (1995).

Abstract

Let $\alpha$ be a unit of degree d in an algebraic number field, and assume that $\alpha$ is not a root of unity. We conduct a numerical investigation that suggests that if $\alpha$ has small Mahler measure, there are many values of n for which $1-\alpha^n$ is a unit and also many values of m for which $\Phi_m(\alpha)$ is a unit, where $\Phi_m$ is the m-th cyclotomic polynomial. We prove that the number of such values of n and m is bounded above by $O(d^{\;1+0.7/\log\log d})$, and we describe a construction of Boyd that gives a lower bound of $\Omega(d^{\;0.6/\log\log d})$.

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Joseph H. Silverman. "Exceptional units and numbers of small Mahler measure." Experiment. Math. 4 (1) 69 - 83, 1995.

Information

Published: 1995
First available in Project Euclid: 3 September 2003

zbMATH: 0851.11064
MathSciNet: MR1359419

Subjects:
Primary: 11R27
Secondary: 11J25

Keywords: Mahler measure , unit equation , units

Rights: Copyright © 1995 A K Peters, Ltd.

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Vol.4 • No. 1 • 1995
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