

A228077


Determinant of the (p_n1)/2 X (p_n1)/2 matrix with (i,j)entry being the Legendre symbol ((ji)/p_n), where p_n is the nth 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
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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_n1)/2}*a(n) = a(n).
M. Vsemirnov has proved Robin Chapman's conjecture on the evaluation of the determinant of the (p+1)/2by(p+1)/2 matrix with (i,j)entry (i,j = 0,...,(p1)/2) being the Legendre symbol ((ji)/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

ZhiWei Sun, Table of n, a(n) for n = 2..200
R. C. Chapman, My evil determinant problem, preprint, 2012.
ZhiWei Sun, On some determinants with Legendre symbol entries, preprint, August 2013.
M. Vseminov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436(2012), 41014106.
M. Vseminov, On R. Chapman's "evil determinant": case p == 1 (mod 4), Acta Arith. 159(2013), 331344.


EXAMPLE

a(2) = 0 since the Legendre symbol ((11)/3) is zero.


MATHEMATICA

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


CROSSREFS

Cf. A226163, A227609, A227968, A227971, A228005.
Cf. A094049.
Sequence in context: A320606 A058177 A204619 * A204170 A283784 A281563
Adjacent sequences: A228074 A228075 A228076 * A228078 A228079 A228080


KEYWORD

sign


AUTHOR

ZhiWei Sun, Aug 09 2013


STATUS

approved



