Values of the Euler Function in Various Sequences

Abstract

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n) r = λ(n) s , where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p−1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f (m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree deg f > 1.

Complexity of inverting the Euler function

“…The congruence condition modulo 8 ensures that (4) holds; the congruence modulo 3 forces the first part of (5) to hold; and the condition modulo 5 forces the second part of (5) to hold. Finally, the condition (6)…”

“…Many aspects of the Euler function, such as growth, distribution in arithmetic progressions and in other special sets of elements, and many other similar questions, have extensively been studied in the literature; see [5,6,10,13,14,15,21,22] and references therein. Nevertheless, the questions considered here seem to be new and have never been studied.…”

Compositions with the euler and carmichael functions

“…For a variety of other results with a similar flavor, we refer the reader to [3,4,5,6,8,10,12,13,21,22,23,24,26,33,34,40,42,45,49] and the references contained therein.…”