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On the Iteration of a Function Related to Euler's φ-Function

  • Joshua Harrington and Lenny Jones
Published/Copyright: November 5, 2010
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Integers
From the journal Volume 10 Issue 5

Abstract

A unit x in a commutative ring R with identity is called exceptional if 1 − x is also a unit in R. For any integer n ≥ 2, define φe(n) to be the number of exceptional units in the ring of integers modulo n. Following work of Shapiro, Mills, Catlin and Noe on iterations of Euler's φ-function, we develop analogous results on iterations of the function φe, when restricted to a particular subset of the positive integers.

Keywords.: Exceptional units; units; the ring of integers modulo n; Euler's φ-function
Received: 2009-02-23
Revised: 2010-03-29
Accepted: 2010-05-12
Published Online: 2010-11-05
Published in Print: 2010-November

© de Gruyter 2010

Articles in the same Issue

  1. Some Divisibility Properties of Binomial Coefficients and the Converse of Wolstenholme's Theorem
  2. On the Iteration of a Function Related to Euler's φ-Function
  3. An Explicit Evaluation of the Gosper Sum
  4. On the Frobenius Problem for {ak, ak + 1, ak + a, . . . , ak + ak−1}
  5. Generalizing the Combinatorics of Binomial Coefficients via -Nomials
  6. Finding Almost Squares V
  7. On Relatively Prime Subsets and Supersets
  8. Reformed Permutations in Mousetrap and Its Generalizations
  9. On Ternary Inclusion-Exclusion Polynomials
  10. Sums and Products of Distinct Sets and Distinct Elements in ℂ
https://doi.org/10.1515/integ.2010.042
Keywords for this article
Exceptional units; units; the ring of integers modulo n; Euler's φ-function

Articles in the same Issue

  1. Some Divisibility Properties of Binomial Coefficients and the Converse of Wolstenholme's Theorem
  2. On the Iteration of a Function Related to Euler's φ-Function
  3. An Explicit Evaluation of the Gosper Sum
  4. On the Frobenius Problem for {ak, ak + 1, ak + a, . . . , ak + ak−1}
  5. Generalizing the Combinatorics of Binomial Coefficients via -Nomials
  6. Finding Almost Squares V
  7. On Relatively Prime Subsets and Supersets
  8. Reformed Permutations in Mousetrap and Its Generalizations
  9. On Ternary Inclusion-Exclusion Polynomials
  10. Sums and Products of Distinct Sets and Distinct Elements in ℂ
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