Two remarks on iterates of Euler’s totient function

Pollack, Paul

doi:10.1007/s00013-011-0314-6

Cited by 3 publications

(4 citation statements)

References 13 publications

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“…where N (x 1 ) is the minimal integer n for which x n = 1 (see also [17]) and it has been conjectured by Erdős et al [4] that N (x 1 ) ∼ α log x 1 as x 1 → ∞, for some α ∈ R. It is known that the understanding of the multiplicative structure of ϕ and its iterates is in some sense equivalent to the study of the behavior of the integers of the form p − 1, where p is prime. See also [8,10,14,18] for related works.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the iterates of shifted Euler's function

Leonetti¹,

Luca²

2023

Preprint

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Let ϕ be the Euler's function and fix an integer k ≥ 0. We show that, for every initial value x 1 ≥ 1, the sequence of positive integers (xn) n≥1 defined by x n+1 = ϕ(xn) + k for all n ≥ 1 is eventually periodic. Similarly, for every initial value x 1 , x 2 ≥ 1, the sequence of positive integers (xn) n≥1 defined by x n+2 = ϕ(x n+1 )+ϕ(xn)+k for all n ≥ 1 is eventually periodic, provided that k is even.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the iterates of shifted Euler's function

Leonetti¹,

Luca²

2023

Preprint

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“…where N(x 1 ) is the minimal integer n for which x n = 1 (see also [17]) and it has been conjectured by Erdős et al [4] that N(x 1 ) ∼ α log x 1 as x 1 → ∞, for some α ∈ R. It is known that the understanding of the multiplicative structure of ϕ and its iterates is, in some sense, equivalent to the study of the behaviour of the integers of the form p − 1, where p is a prime. See also [8,10,14,18] for related work. However, if k ≥ 1, then the function f (n) := ϕ(n) + k does not satisfy (1.2): indeed, ϕ(p) = p − 1 for all primes p, and hence f (p) ≥ p. However, it is well known that…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…It is known that the understanding of the multiplicative structure of and its iterates is, in some sense, equivalent to the study of the behaviour of the integers of the form , where p is a prime. See also [8, 10, 14, 18] for related work.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Iterates of the Shifted Euler’s Function

Leonetti

Luca²

2023

Bull. Aust. Math. Soc.

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Let $\varphi $ be Euler’s function and fix an integer $k\ge 0$ . We show that for every initial value $x_1\ge 1$ , the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for all initial values $x_1,x_2\ge 1$ , the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi (x_{n+1})+\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that k is even.

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“…By applying [13,Theorem 1.1], we are able to obtain the following refined version of [8,Theorem 2.3] which provides the explicit constant 6 π 2 e γ ≈ 0.341326: Theorem 1.2. For large x, we have…”

Section: Introductionmentioning

confidence: 99%

On the Second Moment Estimate Involving the $λ$-Primitive Roots Modulo $n$

Kim¹

2016

Preprint

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Artin's Conjecture on Primitive Roots states that a non-square non unit integer a is a primitive root modulo p for positive proportion of p. This conjecture remains open, but on average, there are many results due to P. J. Stephens (see [14], also [15]). There is a natural generalization of the conjecture for composite moduli. We can consider a as the primitive root modulo (Z/nZ) * if a is an element of maximal exponent in the group. The behavior is more complex for composite moduli, and the corresponding average results are provided by S. Li and C. Pomerance (see [8], [9], and [10]), and recently by the author (see [6]). P. J. Stephens included the second moment results in his work, but for composite moduli, there were no such results previously. We prove that the corresponding second moment result in this case.

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Two remarks on iterates of Euler’s totient function

Abstract: Let ϕ k denote the kth iterate of Euler's ϕ-function. We study two questions connected with these iterates. First, we determine the average order of ϕ k and 1/ϕ k ; e.g., we show that for each k ≥ 0, n≤x

Cited by 3 publications

References 13 publications

On the iterates of shifted Euler's function

On the iterates of shifted Euler's function

On the Iterates of the Shifted Euler’s Function

On the Second Moment Estimate Involving the $λ$-Primitive Roots Modulo $n$

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