Appendix to the note ``The structure of the set of numbers with the Lehmer property'' (Q1036182)

A Lehmer number is a composite positive integer \(n\) such that \(\varphi(n)\mid n-1\), where \(\varphi(n)\) is the Euler function of \(n\). Generalizing a result of Diaconescu, \textit{S. Hernández Hernández} and the reviewer [Integers 8, No. 1, Article A12, 3 p., electronic only (2008; Zbl 1192.11004)] have recently shown that if \(P(X)\in {\mathbb Z}[X]\) is a monic nonconstant polynomial then there are only finitely many Lehmer numbers satisfying the additional property \(P(\varphi(n))\equiv 0\pmod n\). In the paper under review, the authors extend this result in the following sense. They show that given a sequence of monic nonconstant polynomials \((P_M(X))_M\subset {\mathbb Z}[X]\), then there are at most finitely many positive integers \(n\) such that both conditions \(M\varphi(n)=n-1\) and \(P_M(\varphi(n))\equiv 0\pmod n\) hold with some integer \(M\geq 2\), provided that the sequence of polynomials satisfies a certain technical condition. The proof is elementary.