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Popular values of Euler's function

Published online by Cambridge University Press:  26 February 2010

Carl Pomerance
Carl Pomerance
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602.
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For each natural number m, let N(m) denote the number of integers n with ø(n) = m, where ø denotes Euler's function. There are many interesting problems connected with the function N(m), such as the conjecture of Carmichael that N(m) is never 1 (see [9], for example) and the study of the distribution of the m for which N(m) > 0 (see Erdős and Hall [5]). In this note we shall be concerned with the maximal order of N(m).

Type
Research Article
Information
Mathematika , Volume 27 , Issue 1 , June 1980 , pp. 84 - 89
Copyright
Copyright © University College London 1980

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References

1.Bateman, P. T.. “The distribution of values of the Euler function”, Acta Arith., 21 (1972), 329–345.CrossRefGoogle Scholar
2.de Bruijn, N. G.. “On the number of positive integers ≤x and free of prime factors >y, II”, Nederl. Akad. Wetensch. Proc. Ser. A, 69 = Indag. Math., 38 (1966), 239–247.CrossRefGoogle Scholar
3.Erdös, P.. “On the normal number of prime factors of p – 1 and some other related problems concerning Euler's Ø-function”, Quart. J. Math. (Oxford Ser.), 6 (1935), 205–213.CrossRefGoogle Scholar
4.Erdös, P.. “On pseudoprimes and Carmichael numbers”, Publ. Math. Debrecen, 4 (1956), 201–206.Google Scholar
5.Erdös, P. and Hall, R. R.. “Distinct values of Euler's Ø-function”, Mathematika, 23 (1976), 1–3.CrossRefGoogle Scholar
6.Goldfeld, M.. “On the number of primes p for which p + a has a large prime factor”, Mathematika, 16 (1969), 23–27.Google Scholar
7.Hardy, G. H. and Wright, E. M.. An introduction to the theory of numbers (Fourth Ed., Oxford University Press, London, 1960).Google Scholar
8.Hooley, C.. “On the greatest prime factor of p + a”, Mathematika, 20 (1973), 135–143.Google Scholar
9.Pomerance, C.. “On Carmichael's conjecture”, Proc. Amer. Math. Soc, 43 (1974), 297–298.Google Scholar
10.Pomerance, C., Selfridge, J. L. and Wagstaff, S. S.. “The pseudoprimes to 25 109”, Math. Comp., to appear.Google Scholar
11.Wooldridge, K. R.. “Values taken many times by Euler's phi-function”, Proc. Amer. Math. Soc, 76 (1979), 229–234.Google Scholar