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- W2997221840 abstract "We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $pequiv 1pmod 4$ and integer $anotequiv0pmod p$, we prove that begin{align*}&(-1)^{|{1le k<frac p4: (frac kp)=-1}|}prod_{1le j<kle(p-1)/2}(e^{2pi iaj^2/p}+e^{2pi iak^2/p}) =&begin{cases}1&text{if} pequiv1pmod 8,left(frac apright)varepsilon_p^{-(frac ap)h(p)}&text{if} pequiv5pmod8,end{cases} end{align*} and that begin{align*}&left|left{(j,k): 1le j<klefrac{p-1}2 & {aj^2}_p>{ak^2}_pright}right| &+left|left{(j,k): 1le j<klefrac{p-1}2 & {ak^2-aj^2}_p>frac p2right}right| equiv&left|left{1le k<frac p4: left(frac kpright)=left(frac apright)right}right|pmod2. end{align*} where $(frac{a}p)$ is the Legendre symbol, $varepsilon_p$ and $h(p)$ are the fundamental unit and the class number of the real quadratic field $mathbb Q(sqrt p)$ respectively, and ${x}_p$ is the least nonnegative residue of an integer $x$ modulo $p$. Also, for any prime $pequiv3pmod4$ and $delta=1,2$, we determine $$(-1)^{left|left{(j,k): 1le j<kle(p-1)/2 text{and} {delta T_j}_p>{delta T_k}_pright}right|},$$ where $T_m$ denotes the triangular number $m(m+1)/2$." @default.
- W2997221840 created "2020-01-10" @default.
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- W2997221840 date "2019-07-30" @default.
- W2997221840 modified "2023-10-14" @default.
- W2997221840 title "Proof of some conjectures involving quadratic residues" @default.
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- W2997221840 doi "https://doi.org/10.48550/arxiv.1907.12981" @default.
- W2997221840 hasPublicationYear "2019" @default.
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