A new upper bound for numbers with the Lehmer property and its application to repunit numbers

Abstract

A composite positive integer n has the Lehmer property if ϕ(n) divides n − 1, where ϕ is an Euler totient function. In this note we shall prove that if n has the Lehmer property, then n ≤ 2 2 K − 2 2 K−1 , where K is the number of prime divisors of n. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the setwhere ν 2 (g) denotes the highest power of 2 that divides g, and L ≥ 1 is a fixed real number.

On a conjecture of Deaconescu

“…On the other hand, Pomerance [9] proved that every Lehmer number n is < K 2 K , where K = ω(n). Recently, Burek and Żmija [2] have improved this upper bound to 2…”

A group-theoretical approach to Lehmer's totient problem

“…a number which does not satisfy Lehmer's totient problem) belongs to them. We refer to [14][15][16][17][18][19] for some results in this direction. The aim of this paper is to consider the equation…”