Multiplicative structure of values of the Euler function

WD Banks*, JB Friedlander, C Pomerance, IE Shparlinski

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference proceeding contributionpeer-review

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 less than or equal to x for which the value of the Euler function phi(n) is a perfect square and also for the number of n less than or equal to x such that phi(n) is squarefull. We give similar bounds for the Carmichael function lambda(n).

Original languageEnglish
Title of host publicationHigh primes and misdemeanours
Subtitle of host publicationlectures in honour of the 60th birthday of Hugh Cowie Williams
EditorsA VanDerPoorten, A Stein
Place of PublicationProvidence
PublisherAmerican Mathematical Society
Pages29-47
Number of pages19
ISBN (Print)0821833537
Publication statusPublished - 2004
EventInternational Conference in Number Theory in Honour of Hugh Williams on his 60th Birthday - Banff, Alberta, Canada
Duration: 24 May 200330 May 2003

Publication series

NameFields Institute Communications
PublisherAMER MATHEMATICAL SOC
Volume41
ISSN (Print)1069-5265

Conference

ConferenceInternational Conference in Number Theory in Honour of Hugh Williams on his 60th Birthday
CountryCanada
CityBanff, Alberta
Period24/05/0330/05/03

Keywords

  • LARGE PRIME FACTORS
  • NUMBER

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