A372385 Determinant of the matrix [Jacobi(i - j, 2n + 1)]_{0 < i, j < n}, where Jacobi(a, m) denotes the Jacobi symbol (a/m).

0, 1, 2, 1, 8, 16, 2, 1, -56, 1, 0, 0, 44, 1, 127776, 1067089, 592, 7311616, 22, 1, 0, 1, 0, 98867482624, 132, 1933242748921, 31578700795392, 1, 22108, 0, -6001552839958, 1, -2401383183590221824, 1, 818, 0, 95506686981729056, 1, 0, 1, 30328979503109918400, 208688450689382571638784, 394

Offset: 2

Author

Zhi-Wei Sun, Apr 29 2024

Keywords

• sign

Comments

In a 2003 preprint, R. Chapman conjectured that if 2*n + 1 (with n > 1) is a prime congruent to 3 modulo 4, then a(n) = 1.

Conjecture 1: If p = 2*n + 1 is a prime congruent to 1 modulo 4, then det[x + Jacobi(i - j, p)]{0 < i, j < n} = r_p - 2*s_p + (-1)^{n/2}*x*(2*r_p - p*s_p), and det[x + Jacobi(i - j, p)]{0 <= i, j <= n} = -r_p + (-1)^{n/2}*p*s_p*x, where r_p and s_p are rational numbers such that e_p^{(2 - Jacobi(2, p))*h(p)} = r_p + s_p*sqrt(p), e_p and h(p) are the fundamental unit and the class number of the real quadratic field Q(sqrt(p)), respectively.

Conjecture 2: Suppose that p = 2*n + 1 is a prime greater than 3, and write e_p^{h(p)} = a_p + b_p*sqrt(p) with 2*a_p and 2*b_p integral.

(1) det[x + Jacobi(i + j, p)]_{0 < i, j < n} is (-1)^{n/2}*2^{n - 1}*(b_p - a_p*x) if p == 1 (mod 4), and 2^{n-1}*x if p == 3 (mod 4).

(2) det[x + Jacobi(i + j, p)]_{0 <= i, j < n} is (-1)^{n/2}*2^{n-1}*(2*b_p - a_p + x*(p*b_p - 2*a_p)) if p == 1 (mod 4), and -2^{n - 1}*(2*x + 1) if p == 3 (mod 4).

We have checked both conjectures for n up to 1000.

Examples

			a(3) = 1 since the determinant of the matrix [Jacobi(i-j, 2*3+1)]_{0 < i, j < 3} = [0, -1; 1, 0] has the value 1.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 2..144
• R. Chapman, Determinants of Legendre symbol matrices, preprint, 2003. (Cf. Conjecture 2 of this paper.)
• R. Chapman, Determinants of Legendre symbol matrices, Acta Arith. 115 (2004), 231-244.
• Z.-W. Sun, Conjectural values of some determinants involving Legendre symbols (I), Question 470324 at MathOverflow, April 30, 2024.
• M. Vsemirnov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436 (2012), 4101-4106.
• M. Vsemirnov, On R. Chapman's "evil determinant": case p == 1(mod 4), Acta Arith. 159 (2013), 331-344; see also the arXiv version, arXiv:1108.4031 [math.NT], 2011-2012.

Crossrefs

Cf. A179071, A179072, A372314.

Programs

Mathematica

a[n_]:=a[n]=Det[Table[JacobiSymbol[i-j, 2n+1], {i, 1, n-1}, {j, 1, n-1}]];
tab={}; Do[tab=Append[tab, a[n]], {n, 2, 44}]; Print[tab]

PARI

a(n) = matdet(matrix(n-1, n-1, i, j, kronecker(i-j, 2*n+1))); \\ Michel Marcus, Apr 29 2024

Python

from sympy import Matrix, jacobi_symbol
def A372385(n): return Matrix(n-1,n-1,[jacobi_symbol(i-j,(n<<1)|1) for i in range(n-1) for j in range(n-1)]).det() # Chai Wah Wu, Apr 29 2024