✦ LIBER ✦

Regularity of patterns in the factorization of n!

✍ Scribed by D. Berend; G. Kolesnik

Publisher: Elsevier Science
Year: 2007
Tongue: English
Weight: 141 KB
Volume: 124
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2006.08.010

No coin nor oath required. For personal study only.

✦ Synopsis

Consider the multiplicities e p 1 (n), e p 2 (n), . . . , e p k (n) in which the primes p 1 , p 2 , . . . , p k appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m 1 , m 2 , . . . , m k ) for any fixed integers m 1 , m 2 , . . . , m k , thus improving a result of Luca and Stȃnicȃ [F. Luca, P. Stȃnicȃ, On the prime power factorization of n!, J. Number Theory 102 (2003) 298-305].

To prove the theorem, we obtain a result regarding the joint distribution of several completely q-additive functions, which seems to be of independent interest.

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