Mathematics – Number Theory
Scientific paper
2010-12-17
Mathematics
Number Theory
29 pages
Scientific paper
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\sum_{k=0}^{p-1} (1/{m^k}) \binom{2k}k^3$ modulo ${p^2}$. In particular, we show that $\sum_{k=0}^{{p-1/}2}\binom{2k}k^3\e 0\mod {p^2}$ for $p\e 3,5,6\mod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $P_{[p/4]} (t)\e -\big({-6}/p\big)\sum_{x=0}^{p-1}\big({x^3-{3(3t+5)}2x-9t-7}/p\big)\mod p$ and determine $P_{{p-1}/2}(\sqrt 2), P_{{p-1/}2}({3\sqrt 2/}4), P_{{p-1}/2}(\sqrt {-3}),P_{{p-1}/2}({\sqrt 3}/2), P_{{p-1}/2}(\sqrt {-63}), P_{{p-1}/2}({3\sqrt 7}/8)\mod p$, where $t$ is a rational $p-$integer, $[x]$ is the greatest integer not exceeding $x$ and $(a/p)$ is the Legendre symbol. As consequences we determine $P_{[p/4]}(t)\mod p$ in the cases $t=-5/3,-7/9, 65/63$ and confirm several conjectures of Z.W. Sun on $\sum_{k=0}^{p-1}\binom{4k}{2k}\binom{2k}km^{-k}\mod p$ in the cases $m=48,63,72,128$. We also determine $\sum_{k=0}^{[p/4]}\{(4k)!}/{m^k\cdot k!^4}\mod p$ for $m=-144,-1024,-3969,648$.
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