A Remark on the Factorization of Factorials

Document Type : Original Scientific Paper


Department of Mathematics, University of Zanjan, University Blvd., 45371-38791 Zanjan, I. R. Iran


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.


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