Several generalizations on Wolstenholme Theorem

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 ${{(pt + p - 1)!} \over {(pt)!}}\mathop \sum \limits_{m = 0}^{k - 1} {( - 1)^m}\mathop \sum \limits_{1 \le {i_l} < \cdots < {i_{k - m}} \le p - 1} {{{p^{k - (m + 1)}}} \over {\mathop \prod \limits_{l = 1}^{k - m} (pt + {i_l})}} \equiv 0{\left( {\bmod {p^{k + 1}}} \right)^.}$ (pt+p−1)!(pt)!∑m=0k−1(−1)m∑1≤il<⋯<ik−m≤p−1pk−(m+1)∏l=1k−m(pt+il)≡0(modpk+1). The second generalization is on composite number module: Let 1overa ${1 \over a}$1a be the x in congruent equation ax ≡ 1(mod m)(1 ≤ x < m), if m ≥ 5, then $$\matrix{ {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^2}} } \hfill & \equiv \hfill & {{m \over 6}[2m\varphi (m) + \prod\limits_{p|m} {(1 - p)]{{(\bmod m)}^{\;;}}} } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^3}} } \hfill & \equiv \hfill & {{{{m^2}} \over 4}[m\varphi (m) + \prod\limits_{p|m} {(1 - p)](\bmod m){\;^;}} } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^4}} } \hfill & \equiv \hfill & {{m \over {30}}[6{m^3}\varphi (m) + 10{m^2}\prod\limits_{p|m} {(1 - p) - \prod\limits_{p|m} {(1 - {p^3})](\bmod m){\;^;}} } } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^r}} } \hfill & \equiv \hfill & {{m^r}\sum\limits_{d|m} {\mu (d){{\left( {{m \over d}} \right)}^{ - r}}\sum\limits_{k = 1}^{{m \over d}} {{k^r}(\bmod m){\;^.}} } } \hfill \cr } $$∑(k,m)=1,1≤j≤m(1k)2≡m6[2mφ(m)+∏p|m(1−p)](modm) ;∑(k,m)=1,1≤j≤m(1k)3≡m24[mφ(m)+∏p|m(1−p)](modm) ;∑(k,m)=1,1≤j≤m(1k)4≡m30[6m3φ(m)+10m2∏p|m(1−p)−∏p|m(1−p3)](modm) ;∑(k,m)=1,1≤j≤m(1k)r≡mr∑d|mμ(d)(md)−r∑k=1mdkr(modm) . Where φ(x) is Euler function , μ(x) is Möbius function.

Keywords

• wolstenholme theorem
• congruent
• composite number module
• möbius transform