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28 March 2024
 
  » arxiv » 1202.1237

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Jacobsthal sums, Legendre polynomials and binary quadratic forms
Zhi-Hong Sun ;
Date 6 Feb 2012
AbstractLet $p>3$ 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
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