Mersenne数的前导位
Leading Digits of Mersenne Numbers
摘要：
人们早就知道，诸如$2$和因子的幂之类的序列满足本福德定律;也就是说，这些序列中的前导数字以由$P(d)=\log_{10}(1+1/d)$，$d=1,2,\dots,9$给出的频率出现。本文研究了Mersenne数$M_n=2^{p_n}-1$的前导数，其中$p_n$是第$n$个素数.根据素数分布中已知的不规则性，人们可能会认为$\{M_n\}$的前导数字序列比具有类似增长率的“平滑”序列（例如$\{2^{n\log n}\}$）具有\emph{worse}分布特性。令人惊讶的是，相反的似乎是真实的，事实上，我们目前的数据，基于序列$\{M_n\}$的第一个十亿项，显示梅森数的前导位在许多方面的行为\emph{more regularly}比那些在上述光滑序列。我们状态几个apturtures，这种效果，我们提供了一个启发式的解释所观察到的现象的基础上经典模型的分布素数。
英文摘要：
It has long been known that sequences such as the powers of $2$ and the factorials satisfy Benford's Law; that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we consider the leading digits of the Mersenne numbers $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime. In light of known irregularities in the distribution of primes, one might expect that the leading digit sequence of $\{M_n\}$ has \emph{worse} distribution properties than "smooth" sequences with similar rates of growth, such as $\{2^{n\log n}\}$. Surprisingly, the opposite seems to be the true; indeed, we present data, based on the first billion terms of the sequence $\{M_n\}$, showing that leading digits of Mersenne numbers behave in many respects \emph{more regularly} than those in the above smooth sequences. We state several conjectures to this effect, and we provide an heuristic explanation for the observed phenomena based on classic models for the distribution of primes.
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