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June  2020, 28(2): 589-597. doi: 10.3934/era.2020031

Proof of some conjectures involving quadratic residues

1. 

St. Petersburg Department of Steklov, Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023, St. Petersburg, Russia

2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Zhi-Wei Sun

Received  December 2019 Revised  March 2020 Published  April 2020

Fund Project: The work is supported by the NSFC-RFBR Cooperation and Exchange Program (grants NSFC 11811530072 and RFBR 18-51-53020-GFEN-a). The second author is also supported by the Natural Science Foundation of China (grant no. 11971222)

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. Electronic Research Archive, 2020, 28 (2) : 589-597. doi: 10.3934/era.2020031
References:
[1]

B. C. Berndt and S. Chowla, Zero sums of the Legendre symbol, Nordisk Mat. Tidskr., 22 (1974), 5-8.   Google Scholar

[2]

B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, John Wiley & Sons, Inc., New York, 1998.  Google Scholar

[3]

K. Burde, Eine Verteilungseigenschaft der Legendresymbole, J. Number Theory, 12 (1980), 273-277.  doi: 10.1016/0022-314X(80)90063-3.  Google Scholar

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H. Pan, A remark on Zolotarev's theorem, preprint, arXiv: math/0601026. Google Scholar

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Z.-W. Sun, Quadratic residues and related permutations and identities, Finite Fields Appl., 59 (2019), 246-283.  doi: 10.1016/j.ffa.2019.06.004.  Google Scholar

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K. S. Williams and J. D. Currie, Class numbers and biquadratic reciprocity, Canad. J. Math., 34 (1982), 969-988.  doi: 10.4153/CJM-1982-070-x.  Google Scholar

show all references

References:
[1]

B. C. Berndt and S. Chowla, Zero sums of the Legendre symbol, Nordisk Mat. Tidskr., 22 (1974), 5-8.   Google Scholar

[2]

B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, John Wiley & Sons, Inc., New York, 1998.  Google Scholar

[3]

K. Burde, Eine Verteilungseigenschaft der Legendresymbole, J. Number Theory, 12 (1980), 273-277.  doi: 10.1016/0022-314X(80)90063-3.  Google Scholar

[4]

H. Cohn, Advanced Number Theory, Dover Publ., New York, 1962. Google Scholar

[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.  Google Scholar

[6]

H. Pan, A remark on Zolotarev's theorem, preprint, arXiv: math/0601026. Google Scholar

[7]

Z.-W. Sun, Quadratic residues and related permutations and identities, Finite Fields Appl., 59 (2019), 246-283.  doi: 10.1016/j.ffa.2019.06.004.  Google Scholar

[8]

K. S. Williams and J. D. Currie, Class numbers and biquadratic reciprocity, Canad. J. Math., 34 (1982), 969-988.  doi: 10.4153/CJM-1982-070-x.  Google Scholar

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