On composite \(n\) satisfying a problem of Lehmer
Let \(\varphi(n)\) denote Euler's phi-function and \(S_ M\) the set of all positive integers \(n\) for which \(M\phi(n)=n-1\). \textit{D. H. Lehmer} [Bull. Am. Math. Soc. 38, 745--751 (1932; Zbl 0005.34302)] asked whether the sets \(S_ M\) \((M>1)\) contain any composite numbers. He proved that if \(n\in S_ M\) for some \(M>1\) then \(n\) is odd, squarefree, and \(\omega(n)\geq 7\) where \(\omega(n)\) denotes the number of distinct prime factors of \(n\). \textit{F. Schuh} has shown [Math., Zutphen B 12, 102--107 (1944; Zbl 0028.34401)] that if \(n\in S_ M\), \(M>1\), and \(n\equiv 0 \pmod 3\) then \(M\equiv 1\pmod 3\). Others have shown that if \(n\in S_ M\) for some \(M>1\), then \(\omega(n)\geq 14\), and \(\omega(n)\geq 1850\) if \(n\equiv 0 \pmod 3\). It is now established that if \(n\in S_ M\) with \(M>4\) and \(n\equiv 0 \pmod 3\) then \(\omega(n)\geq 5334\).
scientific article; zbMATH DE number 3954998