Several generalizations on Wolstenholme Theorem

Department of electronic and information engineering, Bozhou University 236800, China

^* Corresponding author: zhuyy@hfuu.edu.cn

Abstract

This paper generalizes Wolstenholme theorem on two aspects. The first generalization is a parameterized form: let p > k + 2, k ≥ 1, ∀t ∈ ℤ, then

$\frac{(pt+p-1)!}{(pt)!}{\displaystyle \sum _{m=0}^{k-1}{(-1)}^m}{\displaystyle \sum _{1\le i_l<\cdots <i_{k-m}\le p-1}\frac{p^{k-(m+1)}}{{\displaystyle \prod _{l=1}^{k-m}(pt+i_l)}}\equiv 0}{\left(\mathrm{mod}{p}^{k+1}\right)}^{.}$

The second generalization is on composite number module:

Let 1overa be the x in congruent equation ax ≡ 1(mod m)(1 ≤ x < m),

if m ≥ 5, then

$\begin{array}{lll}\hfill {\displaystyle \sum _{\begin{array}{l}(k,m)=1,\\ 1\le j\le m\end{array}}{\left(\frac{1}{k}\right)}^2}& \equiv \hfill & \frac{m}{6}[2m\phi (m)+{\displaystyle \prod _{p|m}(1-p)]{(\mathrm{mod}m)}^{;}}\hfill \\ \hfill {\displaystyle \sum _{\begin{array}{l}(k,m)=1,\\ 1\le j\le m\end{array}}{\left(\frac{1}{k}\right)}^3}& \equiv \hfill & \frac{m^2}{4}[m\phi (m)+{\displaystyle \prod _{p|m}(1-p)](\mathrm{mod}m){}^{;}}\hfill \\ \hfill {\displaystyle \sum _{\begin{array}{l}(k,m)=1,\\ 1\le j\le m\end{array}}{\left(\frac{1}{k}\right)}^4}& \equiv \hfill & \frac{m}{30}[6m^3\phi (m)+10m^2{\displaystyle \prod _{p|m}(1-p)-{\displaystyle \prod _{p|m}(1-p^3)](\mathrm{mod}m){}^{;}}}\hfill \\ \hfill {\displaystyle \sum _{\begin{array}{l}(k,m)=1,\\ 1\le j\le m\end{array}}{\left(\frac{1}{k}\right)}^r}& \equiv \hfill & m^r{\displaystyle \sum _{d|m}\mu (d){\left(\frac{m}{d}\right)}^{-r}{\displaystyle \sum _{k=1}^{\frac{m}{d}}{k}^r(\mathrm{mod}m){}^{.}}}\hfill \end{array}$

Where φ(x) is Euler function , μ(x) is Möbius function.