The number of solutions of $\phi(x)=m$


An old conjecture of Sierpiński asserts that for every integer $k\ge 2$ , there is a number $m$ for which the equation $\phi(x) = m$ has exactly $k$ solutions. Here $\phi$ 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 Sierpiński’s conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.


Kevin Ford