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The Number of Solutions of φ (x) = m

Kevin Ford
Annals of Mathematics
Second Series, Vol. 150, No. 1 (Jul., 1999), pp. 283-311
Published by: Annals of Mathematics
DOI: 10.2307/121103
Stable URL: http://www.jstor.org/stable/121103
Page Count: 29
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The Number of Solutions of φ  (x) = m
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Abstract

An old conjecture of Sierpinski asserts that for every integer k ≥ 2, there is a number m for which the equation φ (x) = m has exactly k solutions. Here φ is Euler's totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpinski's conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.

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