Some Determinants Involving Quadratic Residues Modulo Primes

Abstract

In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime p > 3 with p ≡ 3 (mod 4) and a, b ∈ ℤ with p ∤ ab, we prove that

$\det {\left[ {1 + \tan\,\pi \frac{{a{j^2} + b{k^2}}}{p}} \right]_{1 \leqslant j,k \leqslant \tfrac{{p - 1}}{2}}} = \left\{ {\begin{array}{*{20}{c}} { - {2^{\frac{{p - 1}}{2}}}{p^{\frac{{p - 3}}{4}}},}&{if \left( {\frac{{ab}}{p}} \right) = 1,} \\ {{p^{\frac{{p - 3}}{4}}},}&{if \left( {\frac{{ab}}{p}} \right) = - 1,} \end{array}} \right.$

denotes the Legendre symbol. We also pose some conjectures for further research.

Keywords

Determinants / Legendre symbols / quadratic residues modulo primes / the tangent function / 11A15 / 11C20 / 15A15 / 33B10

References

[1] BerndtBC, EvansRJ, WilliamsKSGauss and Jacobi Sums, 1998, New York. John Wiley & Sons, Inc..

[2] GrinbergD, SunZ-W, ZhaoL. Proof of three conjectures on determinants related to quadratic residues. Linear Multilinear Algebra, 2022, 70(19): 3734-3746.

[3] IrelandK, RosenMA Classical Introduction to Modern Number Theory, 1990, New York. Springer-Verlag. 84

[4] KrattenthalerC. Advanced determinant calculus: a complement. Linear Algebra Appl., 2005, 411: 68-116.

[5] Singer D., A bijective proof of Borchardt’s identity. Electron. J. Combin., 2004, 11(1): Research Paper 48, 16 pp.

[6] StanleyRPEnumerative Combinatorics, Vol. 1, 2012Second EditionCambridge. Cambridge University Press. 49

[7] SunZ-W. On some determinants with Legendre symbol entries. Finite Fields Appl., 2019, 56: 285-307.

[8] SunZ-W. Quadratic residues and related permutations and identities. Finite Fields Appl., 2019, 59: 246-283.

[9] Sun Z.-W., Arithmetic properties of some permanents. 2021, arXiv:2108.07723

[10] SunZ-W. On some determinants involving the tangent functions. Ramanujan J., 2024, 64(2): 309-332.

[11] Wu H.-L., Quadratic residues and related permutations. Finite Fields Appl., 2019, 60: Paper No. 101576, 10 pp.

[12] Wu H.-L., She Y.-F., Wang L.-Y., Cyclotomic matrices and hypergeometric functions over finite fields. Finite Fields Appl., 2022, 82: Paper No. 102054, 15 pp.

RIGHTS & PERMISSIONS

Peking University