On the prime power factorization of \(n!\). II
For a positive integer \(n\) and a prime \(p\), let \(e_p(n)\) be the power of \(p\) in the prime number factorization of \(n\), i.e. \(p^{e_p(n)}|n\) and \(p^{e_p(n)+1}\nmid n\). The present paper deals with the distribution of \(e_p(n!)\) in residue classes. The following results are proved. Theorem 1. For any positive integer \(m\), any prime \(p\) and any \(\varepsilon \in \mathbb{Z}_m\), there are infinitely many positive integers \(n\) such that \(e_p(n!)\equiv \varepsilon\) (mod \(m\)). Theorem 2. For any positive integer \(m\), there exists a constant \(D(m)\) such that, if \(\varepsilon,\delta\in \mathbb{Z}_m\) and \(p,q\) are two distinct primes with max\(\{p,q\}\geq D(m)\), then there exist infinitely many positive integers \(n\) such that \(e_p(n!)\equiv \varepsilon\) (mod \(m\)) and \(e_q(n!)\equiv \delta\) (mod \(m\)). If \(p\nmid m\) and \(q\nmid m\), these results follow from a theorem of \textit{F. Luca} and \textit{P. Stănică} [J. Number Theory 102, No. 2, 298--305 (2003; Zbl 1049.11092)]. However, their method does not extend to the (more difficult) complementary case which is covered by the above Theorems. The paper concludes with several interesting related open problems.
scientific article; zbMATH DE number 5526758