论文标题
涉及Apéry数字的一致性
On congruences involving Apéry numbers
论文作者
论文摘要
在本文中，我们主要建立了涉及Apéry数字的总和，该总和由Z.-W。太阳。即，对于任何Prime $ p> 3 $和阳性奇数$ M $ c_mp \ left（\ frac {p} {3} \ right）\ pmod {p^3}，$$其中$ a_k = \ sum_ {j = 0}^{k} \ binom {k} $（\ frac {\ cdot} {p}）$是legendre符号。
In this paper, we mainly establish a congruence for a sum involving Apéry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime $p>3$ and positive odd integer $m$, we prove that there is a $p$-adic integer $c_m$ only depending on $m$ such that $$\sum_{k=0}^{p-1}(2k+1)^{m}(-1)^kA_k\equiv c_mp\left(\frac{p}{3}\right)\pmod{p^3},$$ where $A_k=\sum_{j=0}^{k}\binom{k}{j}^2\binom{k+j}{j}^2$ is the Apéry number and $(\frac{\cdot}{p})$ is the Legendre symbol.
