@inproceedings{1996407736894bb191f28d067e6f50e1,

title = "Multiplicative structure of values of the Euler function",

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).",

keywords = "LARGE PRIME FACTORS, NUMBER",

author = "WD Banks and JB Friedlander and C Pomerance and IE Shparlinski",

year = "2004",

language = "English",

isbn = "0821833537",

series = "Fields Institute Communications",

publisher = "American Mathematical Society",

pages = "29--47",

editor = "A VanDerPoorten and A Stein",

booktitle = "High primes and misdemeanours",

address = "United States",

note = "International Conference in Number Theory in Honour of Hugh Williams on his 60th Birthday ; Conference date: 24-05-2003 Through 30-05-2003",

}