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 $.



    加载中


    [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.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1146) PDF downloads(192) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog