On the prime power factorization of n!
| dc.creator | Luca, Florian | |
| dc.creator | Stanica, Pantelimon | |
| dc.date | 2003-04-19 | |
| dc.date.accessioned | 2026-07-07T04:57:04Z | |
| dc.date.available | 2026-07-07T04:57:04Z | |
| dc.description | In this paper we prove two results. The first theorem uses a paper of Kim \cite{K} to show that for fixed primes $p_1,...,p_k$, and for fixed integers $m_1,...,m_k$, with $p_i\not|m_i$, the numbers $(e_{p_1}(n),...,e_{p_k}(n))$ are uniformly distributed modulo $(m_1,...,m_k)$, where $e_p(n)$ is the order of the prime $p$ in the factorization of $n!$. That implies one of Sander's conjecture from \cite{S}, for any set of odd primes. Berend \cite{B} asks to find the fastest growing function $f(x)$ so that for large $x$ and any given finite sequence $ε_i\in \{0,1\}, i\le f(x)$, there exists $n<x$ such that the congruences $e_{p_i}(n)\equiv ε_i\pmod 2$ hold for all $i\le f(x)$. Here, $p_i$ is the $i$th prime number. In our second result, we are able to show that $f(x)$ can be taken to be at least $c_1 (\log x/(\log\log x)^6)^{1/9}$, with some absolute constant $c_1$, provided that only the first odd prime numbers are involved. | |
| dc.description | 7 pages; accepted Journal of Number Theory | |
| dc.identifier | https://arxiv.org/abs/math/0304272 | |
| dc.identifier | http://arxiv.org/abs/math/0304272 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67144 | |
| dc.subject | Number Theory | |
| dc.subject | 11N25; 11B50 | |
| dc.title | On the prime power factorization of n! | |
| dc.type | text |