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On a problem of Chen and Liu concerning the prime power factorization of $ n!$

Authors: Johannes F. Morgenbesser and Thomas Stoll
Journal: Proc. Amer. Math. Soc. 141 (2013), 2289-2297
MSC (2010): Primary 11N25; Secondary 11A63, 11B50, 11L07, 11N37
Published electronically: March 29, 2013
MathSciNet review: 3043010
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Abstract: For a fixed prime $ p$, let $ e_p(n!)$ denote the order of $ p$ in the prime factorization of $ n!$. Chen and Liu (2007) asked whether for any fixed $ m$, one has $ \{e_p(n^2!) \bmod m:\; n\in \mathbb{Z}\}=\mathbb{Z}_m$ and $ \{e_p(q!) \bmod m:\; q$$ \mbox { prime}\}=\mathbb{Z}_m$. We answer these two questions and show asymptotic formulas for $ \char93 \{n<x: n \equiv a \bmod d,\; e_p(n^2!)\equiv r \bmod m\}$ and $ \char93 \{q<x: q$$ \mbox { prime}, q \equiv a \bmod d,\; e_p(q!)\equiv r \bmod m\}$. Furthermore, we show that for each $ h\ge 3$, we have $ \char93 \{n<x: n \equiv a \bmod d,\; e_p(n^h!)\equiv r \bmod m\} \gg x^{4/(3h+1)}$.

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Johannes F. Morgenbesser
Affiliation: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8–10, A–1040 Wien, Austria – and – Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria

Thomas Stoll
Affiliation: Institut de Mathématiques de Luminy, Université d’Aix-Marseille, 13288 Marseille Cedex 9, France

Keywords: Prime power factorization, $p$-adic valuation, sum of digits, congruences, squares, primes
Received by editor(s): October 21, 2011
Published electronically: March 29, 2013
Additional Notes: The first author was supported by the Austrian Science Foundation FWF, grants S9604 and P21209.
This research was supported by the Agence Nationale de la Recherche, grant ANR-10-BLAN 0103 MUNUM
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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