Wolstenholme定理：一百五十年来的推广与扩展（1862- 2012）
Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
摘要：
Wolstenholme在1862年证明，任何总理$p\ge 5$的分子的分数$$ 1+\frac 12 +\frac 13+...+\frac{1}{p-1} $$书面减少形式是整除的$p^2$，$(2)$和分子的分数$$ 1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2} $$书面减少形式是整除的$p$。上述的第一个同余，即所谓的{ \it Wolstenholme定理}，是组合数论中的一个基本同余。本文共分11节，对Wolstenholme型同余及其相关问题进行了历史回顾。也就是说，我们提出和比较几个推广和推广Wolstenholme的定理在过去的一百五十年。特别是，我们提出了70多个变化和推广这个定理，包括同余Wolstenholme素数。这些一致性在这里讨论了33注释。本文参考文献106篇，包括13本教科书和专著，89篇论文，3个问题和Sloane的联机Enc。的序列。本文引用了这些文献中的一些结果作为某些Wolstenholme型同余的推广，但没有对相关同余的论述。这里引用的总数量为189。
英文摘要：
In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$ 1+\frac 12 +\frac 13+...+\frac{1}{p-1} $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$ 1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2} $$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. Namely, we present and compare several generalizations and extensions of Wolstenholme's theorem obtained in the last hundred and fifty years. In particular, we present more than 70 variations and generalizations of this theorem including congruences for Wolstenholme primes. These congruences are discussed here by 33 remarks. The Bibliography of this article contains 106 references consisting of 13 textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of Integer Sequences. In this article, some results of these references are cited as generalizations of certain Wolstenholme's type congruences, but without the expositions of related congruences. The total number of citations given here is 189.
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