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تسجيل مستخدم جديدThree supercongruences for Apery numbers or Franel numbers
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Yong Zhang
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The Apery numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=sum_{k=0}^{n}{binom{n+k}{2k}}^2{binom{2k}{k}}^2 {rm and } f_n=sum_{k=0}^{n}{binom{n}{k}}^3(n=0, 1, cdots,).$$ In this paper, we prove three supercongruences for Apery numbers or Franel numbers conjectured by Z.-W. Sun. Let $pgeq 5$ be a prime and let $nin mathbb{Z}^{+}$. We show that begin{align} otag frac{1}{n}bigg(sum_{k=0}^{pn-1}(2k+1)A_k-psum_{k=0}^{n-1}(2k+1)A_kbigg)equiv0pmod{p^{4+3 u_p(n)}} end{align} and begin{align} otag frac{1}{n^3}bigg(sum_{k=0}^{pn-1}(2k+1)^3A_k-p^3sum_{k=0}^{n-1}(2k+1)^3A_kbigg)equiv0pmod{p^{6+3 u_p(n)}}, end{align} where $ u_p(n)$ denotes the $p$-adic order of $n$. Also, for any prime $p$ we have begin{align} otag frac{1}{n^3}bigg(sum_{k=0}^{pn-1}(3k+2)(-1)^kf_k-p^2sum_{k=0}^{n-1}(3k+2)(-1)^kf_kbigg)equiv0pmod{p^{3}}. end{align}
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