A372314 Determinant of the matrix [Jacobi(i^2 + 3*i*j + 2*j^2, 2*n + 1)]_{1 < i, j < 2*n}, where Jacobi(a, m) denotes the Jacobi symbol (a / m).

1, 3, 0, 125, -1215, 0, 0, 9126441, 0, -187590821, 0, 0, 20686753425, 0, 0, 0, 9224101117395305225, 0, 881852208012283730302080, 624391710361368134976, 0, -3428714319207136609529065, 0, 0, 3878246452353765171209988566241, 0, 0, 4308304210666498856284267223158421

Offset: 2

Author

Zhi-Wei Sun, Apr 27 2024

Keywords

• sign

Comments

Conjecture 1: Let n be any positive integer.

(i) If a(2*n) is nonzero, then 4*n + 1 is a sum of two squares.

(ii) a(2*n + 1) is divisible by phi(4*n + 3)/2, where phi is Euler's totient function. If n is even, then a(2*n + 1)/(phi(4*n + 3)/2) is a square. This has been verified for n = 2..1000.

For any odd integer n > 3 and integers c and d, we introduce the notation: {c,d}n = det[Jacobi(i^2 + c*i*j + d*j^2, n)]{1 < i, j < n-1}.

The following conjecture is similar to Conjecture 1.

Conjecture 2: (1) {2, 2}_p = 0 for any prime p == 13,19 (mod 24), and {2, 2}_p == 0 (mod p) for any prime p == 17,23 (mod 24).

(2) If n == 5 (mod 8), then {4, 2}_n = 0. If n == 5 (mod 12), then {3, 3}_n = 0.

(3) If n == 5 (mod 12) and n is a sum of two squares, then {10, 9}_n = 0. Also, {10, 9}_p == 0 (mod p) for any prime p == 11 (mod 12).

(4) {8, 18}_p == 0 (mod p^2) for any prime p == 19 (mod 24), and {8,18}_p == 0 (mod p) for any prime p == 23 (mod 24). If n == 13,17 (mod 24) and n is a sum of two squares, then {8, 18}_n = 0.

We have verified Conjecture 2 for p or n smaller than 2000.

Examples

			a(2) = 1 since the determinant of the matrix [Jacobi(i^2 + 3*i*j + 2*j^2, 5)]_{1 < i, j < 2*2} = [1,0; 0,1] is 1.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 2..73
• D. Krachun, F. Petrov, Z.-W. Sun and M. Vsemirnov, On some determinants involving Jacobi symbols, Finite Fields Appl. 64 (2010), Article 101672.
• Z.-W. Sun, On some determinants with Legendre symbol entries, Finite Fields Appl. 56 (2019), 285-307.

Crossrefs

Cf. A000010, A000040, A001481.

Programs

Mathematica

a[n_]:=a[n]=Det[Table[JacobiSymbol[i^2+3*i*j+2*j^2,2n+1],{i,2,2n-1},{j,2,2n-1}]];
tab={};Do[tab=Append[tab,a[n]],{n,2,29}];Print[tab]

PARI

f(i,j) = i^2 + 3*i*j + 2*j^2;
a(n) = matdet(matrix(2*n-2, 2*n-2, i, j, kronecker(f(i+1,j+1), 2*n+1)));
vector(25, n, a(n+1)) \\ Michel Marcus, Apr 27 2024