论文标题
涉及Motzkin数字和中央三项系数的超企业
Supercongruences involving Motzkin numbers and central trinomial coefficients
论文作者
论文摘要
令$ m_n $和$ t_n $分别表示$ n $ th motzkin编号和$ n $ th中央三项系数。我们证明，对于任何prime $ p \ ge 5 $，\ begin {align*}＆\ sum_ {k = 0}^{p-1} m_k^2 \ equiv \ equiv \ left（\ frac {p} {3} {3} {3} {3} {3} \ right） ＆\ sum_ {k = 0}^{p-1} km_k^2 \ equiv \ left（\ frac {p} {3} {3} \ right）\ left（9p-1 \ right）\ pmod {p^2} \ frac {4} {3} \ left（\ frac {p} {3} \右）+\ frac {p} {6} {6} {6} \ left（1-9 \ left（\ frac {p} {p} {p} {3} {3} {3} {3} {3} {3} {3} {3} \ right）是Legendre符号。这些结果证实了Z.-W。的三个12岁的超级企业猜想。太阳。
Let $M_n$ and $T_n$ denote the $n$th Motzkin number and the $n$th central trinomial coefficient respectively. We prove that for any prime $p\ge 5$, \begin{align*} &\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\ &\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\ &\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2}, \end{align*} where $\left(-\right)$ is the Legendre symbol. These results confirm three 12-year-old supercongruence conjectures of Z.-W. Sun.
