What is Lehmer's number?

Eriko Hironaka

Lehmer's number was first discovered by Lehmer in 1933, as part of a search for formulas that would produce large primes with high probability. It turns out to be a number that also answers questions about "minimality" and "complexity" of objects from different fields of mathematics from number theory to geometry. This paper describes some of the appearances of Lehmer's number and how they reveal underlying relations between objects as seemingly diverse as algebraic numbers, knots and links, mapping classes, graphs and Coxeter systems.