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- W2950783457 abstract "Let $p$ be an odd prime and let $mnotequiv 0pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $sum_{k=0}^{[p/4]}binom{4k}{2k}frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,binBbb Z$. When $pnmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{frac{q-1}4}equiv (frac ab)^rpmod q$ if and only if $$sum_{k=0}^{[p/4]}binom{4k}{2k}Big(frac{a^2}{16q}Big)^kequiv (-1)^{frac{p^2-1}8a+frac{p-1}2cdot frac{q-1}4}Big(frac pqBig) Big(frac abBig)^rpmod p,$$ where $(frac pq)$ is the Legendre symbol. We also establish congruences for $sum_{k=0}^{[p/4]}binom{4k}{2k}frac 1{m^k}pmod p$ in the cases $m=17,18,20,32,52,80,272$." @default.
- W2950783457 created "2019-06-27" @default.
- W2950783457 creator A5069138836 @default.
- W2950783457 date "2013-11-25" @default.
- W2950783457 modified "2023-10-16" @default.
- W2950783457 title "Quartic residues and sums involving $binom{4k}{2k}$" @default.
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- W2950783457 doi "https://doi.org/10.48550/arxiv.1311.6364" @default.
- W2950783457 hasPublicationYear "2013" @default.
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