A Remark on the Factorization of Factorials
نویسندگان
1 Department of Mathematics, University of Zanjan, University Blvd., 45371-38791 Zanjan, I. R. Iran
2 Department of Mathematics, University of Zanjan, University Blvd., 45371-38791 Zanjan, I. R. Iran
doi
10.22052/mir.2021.240348.1254چکیده
The subject of this paper is to study distribution of the prime factors p and their exponents, which we denote by vp (n!), in standard factorization of n! into primes. We show that for each θ > 0 the primes p not exceeding nθ eventually assume almost all value of the sum ∑p⩽nθ vp(n!). Also, we introduce the notion of θ-truncated factorial, defined by n!θ =∏p⩽nθ pvp (n!) and we show that the growth of log n!1/2 is almost half of growth of log n!1.
