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

2011/10/21 by Johannes F. Morgenbesser, Thomas Stoll, Morgenbesser, Johannes F. +2
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Finite Group Theory Research #graph theory and CDMA systems #math.NT #msc:11A63 #msc:11B50 #msc:11L07 #msc:11N25 #msc:11N37

paper · pdf · doi:10.48550/arxiv.1110.4814

9 pages

arxiv created 2011/10/21 · arxiv updated 2011/10/24

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(n²!) \bmod m: n∈ℤ\=ℤ_m and \e_p(q!) \bmod m: q prime\=ℤ_m. We answer these two questions and show asymptotic formulas for # \n<x: n ≡ a \bmod d, e_p(n²!)≡ r \bmod m\ and # \q<x: q prime, q ≡ a \bmod d, e_p(q!)≡ r \bmod m\. Furthermore, we show that for each h≥ 3, we have \n<x: n ≡ a \bmod d, e_p(n^h!)≡ r \bmod m\ ≫ x^4/(3h+1).

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