A228077 Determinant of the (p_n-1)/2 X (p_n-1)/2 matrix with (i,j)-entry being the Legendre symbol ((j-i)/p_n), where p_n is the n-th prime.

0, -1, 0, 0, -5, 1, 0, 0, -13, 0, -145, 5, 0, 0, -25, 0, -3805, 0, 0, 125, 0, 0, 53, 569, -401, 0, 0, -851525, 73, 0, 0, 149, 0, -9305, 0, -385645, 0, 0, -85, 0, -82596761, 0, 126985, -785, 0, 0, 0, 0, -1321693313, 1517, 0, 4574225, 0, 1025, 0, -134485, 0, -535979945, 63445, 0, -145, 0, 0, 7170685, -19805, 0, 55335641, 0, -167273125693, 3793, 0, 0, -27765559705, 0, 0, -427316305, -1027776565, 2564801, 5534176685

Offset: 2

Author

Zhi-Wei Sun, Aug 09 2013

Keywords

• sign

Comments

Conjecture: In the case p_n == 1 (mod 4), (2/p_n)*a(n) is a positive odd integer whose prime factors are all congruent to 1 modulo 4, and moreover for some integer b(n) we have b(n) + (2/p_n)*a(n)*sqrt(p_n) = e(p_n)^{(2-(2/p_n))h(p_n)}, where e(p_n) and h(p_n) are the fundamental unit and the class number of the real quadratic field Q(sqrt(p_n)) respectively.

Note that a(n) = 0 when p_n == 3 (mod 4), this is because the transpose of the determinant a(n) coincides with (-1/p_n)^{(p_n-1)/2}*a(n) = -a(n).

M. Vsemirnov has proved Robin Chapman's conjecture on the evaluation of the determinant of the (p+1)/2-by-(p+1)/2 matrix with (i,j)-entry (i,j = 0,...,(p-1)/2) being the Legendre symbol ((j-i)/p), where p is an odd prime.

On Aug 14 2013, Robin Chapman informed the author that he first made the conjecture about the exact value of a(n) in a private manuscript dated Aug 05 2003.

Examples

			a(2) = 0 since the Legendre symbol ((1-1)/3) is zero.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 2..200
• R. C. Chapman, My evil determinant problem, preprint, 2012.
• Zhi-Wei Sun, On some determinants with Legendre symbol entries, arXiv:1308.2900 [math.NT], 2010-2013; Finite Fields Appl. 56(2019), 285-307.
• Maxim Vsemirnov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436(2012), 4101-4106.
• Maxim Vsemirnov, On R. Chapman's "evil determinant": case p == 1 (mod 4), Acta Arith. 159(2013), 331-344.

Crossrefs

Cf. A226163, A227609, A227968, A227971, A228005.

Cf. A094049.

Programs

Mathematica

a[n_]:=Det[Table[JacobiSymbol[j-i,Prime[n]],{i,1,(Prime[n]-1)/2},{j,1,(Prime[n]-1)/2}]]
Table[a[n],{n,2,20}]

PARI

a(n) = my(p=prime(n)); matdet(matrix((p-1)/2, (p-1)/2, i, j, i--; j--; kronecker(i-j, p))); \\ Michel Marcus, Aug 25 2021