Multiplicative structure of values of the Euler function

Abstract

Abstract. We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain "smoothing" effect on its integer arguments, our results show that, in fact, most values produced by the Euler function are not smooth. We apply our results to study the distribution of "strong primes", which are commonly encountered in cryptography.We also consider the problem of obtaining upper and lower bounds for the number of positive integers n ≤ x for which the value of the Euler function ϕ(n) is a perfect square and also for the number of n ≤ x such that ϕ(n) is squarefull. We give similar bounds for the Carmichael function λ(n).

Values of the Euler Function in Various Sequences

“…We recall that, for any positive integer n, ϕ(n) is the cardinality of the multiplicative group U n = (Z/nZ) × , while λ(n) is the maximal order of any element in U n . There exists an extensive literature in which the distributional and arithmetical properties of ϕ(n) and λ(n) have been studied (for example, see [1,3,4,5,6,8,9,12,13,14,15,16,17,24,26,28,29]). Here, we list a few examples of properties and interrelations between ϕ(n), λ(n) and n that have been investigated in those works:…”

“…• Positive integers n such that ϕ(n) is smooth, and those for which ϕ(n) is a perfect square, have been studied in [3].…”

“…where k ≥ 2 is an integer; for example, ϕ(1729) = λ(1729) 2 , ϕ(666) 2 = λ(666) 3 , ϕ(768) 3 = λ(768) 4 , . .…”

“…As we have remarked, in the special case f (X) = X 2 , this problem (and other similar ones) has been studied in [3]. However, the underlying method of that paper cannot be extended to work for general polynomials f .…”

Triangular numbers whose sum of divisors is also triangular

“…Assume further that f factors completely over Q and has only simple roots. Then there exists a positive constant c := c(f, g) such that (1) max{#F f,g (x), #S f,g (x)} ≤ c x (ln x) 1/10 for all x > e. Here, ln x denotes the natural logarithm of x.…”