arXiv math/03042722003-04-19math.NT

关于n的素幂分解！

On the prime power factorization of n!

Florian Luca, Pantelimon Stanica

备注：: 7 pages; accepted Journal of Number Theory

摘要：

在本文中，我们证明了两个结果。第一个定理使用金\cite{K}的一份文件，以显示，对于固定素数$p_1,...,p_k$，并为固定整数$m_1,...,m_k$，与$p_i\not|m_i$，数字$(e_{p_1}(n),...,e_{p_k}(n))$是均匀分布模$(m_1,...,m_k)$，其中$e_p(n)$是阶的素数$p$的因式分解$n!$。这意味着桑德的猜想之一，从PDMATHAMV，对任何一组奇素数。Berend \cite{B}要求找到增长最快的函数$f(x)$，使得对于大的$x$和任何给定的有限序列$ε_i\in \{0,1\}, i\le f(x)$，存在$n<x$，使得同余$e_{p_i}(n)\equiv ε_i\pmod 2$对所有$i\le f(x)$成立。这里，$p_i$是$i$的第一个素数。在我们的第二个结果中，我们能够证明$f(x)$至少可以被认为是$c_1 (\log x/(\log\log x)^6)^{1/9}$，具有某个绝对常数$c_1$，只要只涉及第一个奇素数。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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