### Electronic Research Archive

2020, Issue 2: 589-597. doi: 10.3934/era.2020031

# Proof of some conjectures involving quadratic residues

• Received: 01 December 2019 Revised: 01 March 2020
• Primary: 11A15, 05A05; Secondary: 11R11, 33B10

• We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\ ({\rm{mod}}\ 4)$ and integer $a\not\equiv0\ ({\rm{mod}}\ p)$, we prove that

$(-1)^{|\{1 \leq k<\frac p4:\ (\frac kp) = -1\}|}\prod\limits_{1 \leq j<k \leq (p-1)/2}(e^{2\pi iaj^2/p}+e^{2\pi iak^2/p}) \\ = \begin{cases}1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ p\equiv1\ ({\rm{mod}}\ 8), \\ \left(\frac ap\right)\varepsilon_p^{-(\frac ap)h(p)}\ \ \text{if}\ p\equiv5\ ({\rm{mod}}\ 8), \end{cases}$

and that

$\begin{array}{*{35}{l}}\left|\left\{(j, k):\ 1 \leq j<k \leq \frac{p-1}2\ \ \{aj^2\}_p>\{ak^2\}_p\right\}\right| \\ +\left|\left\{(j, k):\ 1 \leq j<k \leq \frac{p-1}2\ \ \{ak^2-aj^2\}_p>\frac p2\right\}\right| \\ \equiv \left|\left\{1 \leq k<\frac p4:\ \left(\frac kp\right) = \left(\frac ap\right)\right\}\right|\ ({\rm{mod}}\ 2), \end{array}$

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 $p\equiv3\ ({\rm{mod}}\ 4)$ and ${\delta} = 1, 2$, we determine

$(-1)^{\left|\left\{(j, k): \ 1 \leq j<k \leq (p-1)/2\ \text{and}\ \{{\delta} T_j\}_p>\{{\delta} T_k\}_p\right\}\right|},$

where $T_m$ denotes the triangular number $m(m+1)/2$.

Citation: Fedor Petrov, Zhi-Wei Sun. Proof of some conjectures involving quadratic residues[J]. Electronic Research Archive, 2020, 28(2): 589-597. doi: 10.3934/era.2020031

### Related Papers:

• We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\ ({\rm{mod}}\ 4)$ and integer $a\not\equiv0\ ({\rm{mod}}\ p)$, we prove that

$(-1)^{|\{1 \leq k<\frac p4:\ (\frac kp) = -1\}|}\prod\limits_{1 \leq j<k \leq (p-1)/2}(e^{2\pi iaj^2/p}+e^{2\pi iak^2/p}) \\ = \begin{cases}1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ p\equiv1\ ({\rm{mod}}\ 8), \\ \left(\frac ap\right)\varepsilon_p^{-(\frac ap)h(p)}\ \ \text{if}\ p\equiv5\ ({\rm{mod}}\ 8), \end{cases}$

and that

$\begin{array}{*{35}{l}}\left|\left\{(j, k):\ 1 \leq j<k \leq \frac{p-1}2\ \ \{aj^2\}_p>\{ak^2\}_p\right\}\right| \\ +\left|\left\{(j, k):\ 1 \leq j<k \leq \frac{p-1}2\ \ \{ak^2-aj^2\}_p>\frac p2\right\}\right| \\ \equiv \left|\left\{1 \leq k<\frac p4:\ \left(\frac kp\right) = \left(\frac ap\right)\right\}\right|\ ({\rm{mod}}\ 2), \end{array}$

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 $p\equiv3\ ({\rm{mod}}\ 4)$ and ${\delta} = 1, 2$, we determine

$(-1)^{\left|\left\{(j, k): \ 1 \leq j<k \leq (p-1)/2\ \text{and}\ \{{\delta} T_j\}_p>\{{\delta} T_k\}_p\right\}\right|},$

where $T_m$ denotes the triangular number $m(m+1)/2$.

 [1] Zero sums of the Legendre symbol. Nordisk Mat. Tidskr. (1974) 22: 5-8. [2] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, John Wiley & Sons, Inc., New York, 1998. [3] Eine Verteilungseigenschaft der Legendresymbole. J. Number Theory (1980) 12: 273-277. [4] H. Cohn, Advanced Number Theory, Dover Publ., New York, 1962. [5] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd Edition, Grad. Texts. Math., 84. Springer, New York, 1990. doi: 10.1007/978-1-4757-2103-4 [6] H. Pan, A remark on Zolotarev's theorem, preprint, arXiv: math/0601026. [7] Quadratic residues and related permutations and identities. Finite Fields Appl. (2019) 59: 246-283. [8] Class numbers and biquadratic reciprocity. Canad. J. Math. (1982) 34: 969-988.
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