Carmichael's ``Empirical Theorem''
If \(n\) is a natural number and \(\varphi\) is Euler's totient function, let \(N(n)\) denote the number of solutions of the equation \(\varphi (x)=n\). More than sixty years ago R. D. Carmichael conjectured that for every \(n\) either \(N(n)=0\) or \(N(n)\geq 2\). This conjecture is equivalent to the assertion that if \(X=\{x: N(\varphi (x))=1\}\) then \(X\) is empty. The present paper discusses the history of Carmichael's conjecture and the progress which has been made in the study of the elements of \(X\). Particular attention is paid to the result obtained by \textit{P. Masai} and \textit{A. Valette} [Boll. Unione Mat. Ital., IV. Ser., A 1, 313--316 (1982; Zbl 0484.10004)] that if \(x\in X\) then \(x>10^{10 000}\). The author points out that a very significant improvement in this lower bound for \(X\) could almost certainly be obtained by making only a minor adjustment in the Masai-Valette method.
scientific article; zbMATH DE number 3937236