|
|
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.
|
|
4
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,5
|
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
a(2) = 0 since the Legendre symbol ((1-1)/3) is zero.
|
|
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}]
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|