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Congruences concerning Legendre polynomials III | | Zhi-Hong Sun
; | | Date: |
20 Dec 2010 | | Abstract: | Let $p>3$ be a prime, and let $m$ be an integer with $p
mid m$. In the paper
we solve some conjectures of Z.W. Sun concerning
$sum_{k=0}^{p-1}frac{(6k)!}{m^k(3k)!k!^3}mod p$, and show that for integers
$m,n$ with $p
mid m$,
$$Big(sum_{x=0}^{p-1}Big(frac{x^3+mx+n}pBig)Big)^2e
Big(frac{-3m}pBig)
sum_{k=0}^{[p/6]}frac{(6k)!}{(3k)!k!^3}Big(frac{4m^3+27n^2}{12^3cdot
4m^3}Big)^kmod p,$$ where $sls ap$ is the Legendre symbol and $[x]$ is the
greatest integer function. | | Source: | arXiv, 1012.4234 | | Services: | Forum | Review | PDF | Favorites | |
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