论文标题
阶乘$ \ pmod p $和模块化映射的平均值
Factorials $\pmod p$ and the average of modular mappings
论文作者
论文摘要
我们知道，$ \ Mathcal {m} = \ {1,2，\ dots，m \} $中的大多数序列，长度$ n $都会想念$ me^{ - λ} $，总数为$ \ {1,2，\ dots，\ dots，m \} $，是$ n/m $ n/m $ to $ n/m $ to $λ$。现在，我们考虑了一个更普遍的情况，即$ \ {1,2，\ dots，m \} $中的数字通过“随机”序列$ f（1），f（2），\ dots，f（n）$来实现k次。我们表明，如果$ n/m \rightarrowλ$，那么限制具有泊松分布，即，$ \ mathcal {m} $中某个数字的序列比例正好获得$ k $ times具有限制$ \ frac $ \ frac {λ^k}^k} {k} {k！} {k！我们猜想这是阶乘映射模型的行为，并提出一些支持论点。
We have known that most sequences in $\mathcal{M}=\{1,2,\dots, M\}$ with length $n$ will miss $Me^{-λ}$ of the total numbers of $\{1,2,\dots,M\}$ as the ratio $n/M$ tends to $λ$. Now we consider a more general case where the numbers in $\{1,2,\dots,M\}$ are achieved exactly k times by a 'random' sequence $f(1), f(2),\dots,f(n)$. We show that if $n/M\rightarrow λ$, then the limit has a Poisson distribution, that is, the proportion of sequences for which some number in $\mathcal{M}$ is achieved exactly $k$ times has the limit $\frac{λ^k}{k!}e^{-λ}$. We conjecture that this is the behavior of the factorial mapping modulo a prime and present a few supporting arguments.
