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Jacobsthal sums, Legendre polynomials and binary quadratic forms | Zhi-Hong Sun
; | Date: |
6 Feb 2012 | Abstract: | Let $p>3$ be a prime and $m,ninBbb Z$ with $p
mid mn$. Built on the work
of Morton, in the paper we prove the uniform congruence:
$$&sum_{x=0}^{p-1}Big(frac{x^3+mx+n}pBig) equiv {-(-3m)^{frac{p-1}4}
sum_{k=0}^{p-1}inom{-frac 1{12}}kinom{-frac 5{12}}k
(frac{4m^3+27n^2}{4m^3})^kpmod p& {if $4mid p-1$,}
frac{2m}{9n}(frac{-3m}p)(-3m)^{frac{p+1}4} sum_{k=0}^{p-1}inom{-frac
1{12}}kinom{-frac 5{12}}k (frac{4m^3+27n^2}{4m^3})^kpmod p& ext{if $4mid
p-3$,}$$ where $(frac ap)$ is the Legendre symbol. We also establish many
congruences for $xpmod p$, where $x$ is given by $p=x^2+dy^2$ or
$4p=x^2+dy^2$, and pose some conjectures on supercongruences modulo $p^2$
concerning binary quadratic forms. | Source: | arXiv, 1202.1237 | Services: | Forum | Review | PDF | Favorites |
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