## Experimental Mathematics

- Experiment. Math.
- Volume 4, Issue 1 (1995), 69-83.

### Exceptional units and numbers of small Mahler measure

#### 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 $\Omega(d^{\;0.6/\log\log d})$.

#### Article information

**Source**

Experiment. Math., Volume 4, Issue 1 (1995), 69-83.

**Dates**

First available in Project Euclid: 3 September 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1062621144

**Mathematical Reviews number (MathSciNet)**

MR1359419

**Zentralblatt MATH identifier**

0851.11064

**Subjects**

Primary: 11R27: Units and factorization

Secondary: 11J25: Diophantine inequalities [See also 11D75]

**Keywords**

units Mahler measure unit equation

#### Citation

Silverman, Joseph H. Exceptional units and numbers of small Mahler measure. Experiment. Math. 4 (1995), no. 1, 69--83. https://projecteuclid.org/euclid.em/1062621144