The number of solutions of phi(x)=m

Abstract

An old conjecture of Sierpinski asserts that for every integer k \ge 2, there is a number m for which the equation ϕ(x)=m has exactly k solutions. Here ϕis 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.