On sparsely totient numbers

David Masser and Peter Man-Kit Shiu

Vol. 121 (1986), No. 2, 407–426

DOI: 10.2140/pjm.1986.121.407

Abstract

Let φ(n) denote Euler’s totient function, defined for n > 1 by

Let F be the set of integers n > 1 with the property that φ(m) > φ(n) whenever m > n. The purpose of this paper is to establish a number of results about the set F. For example, we shall prove that each prime divides all sufficiently large elements of F, each positive integer divides some element of F, and that the ratio of successive elements of F approaches 1.

Mathematical Subject Classification 2000

Primary: 11N64

Secondary: 11A25

Milestones

Received: 13 February 1984

Published: 1 February 1986