A179072 - OEIS

%I #28 May 08 2024 14:51:02

%S -1,-2,0,0,-32,256,0,0,-8192,0,-262144,5242880,0,0,-33554432,0,

%T -2684354560,0,0,8589934592000,0,0,932385860354048,160159261748363264,

%U -1125899906842624,0,0,-225179981368524800,5260204364768739328,0,0

%N Chapman's "evil" determinants II.

%C Determinant of the k-by-k matrix with (i,j)-entry L((i+j)/p), where L(./p) denotes the Legendre symbol modulo p and p = p_n = 2k+1 is the n-th prime.

%C Guy says "Chapman has a number of conjectures which concern the distribution of quadratic residues." One is that if 3 < p_n == 3 (mod 4), then a(n) = 0.

%C It appears that a(n) is even, if p_n == 1 (mod 4).

%C For any odd prime p, (p+1)/2-i+(p+1)/2-j == -(i+j-1) (mod p) and hence we have L(-1/p)*|L((i+j)/p)|_{i,j=1,...,(p-1)/2} = |L((i+j-1)/p)|_{i,j=1,...,(p-1)/2}. Thus the value of a(n) was actually determined in the first reference of R. Chapman. - _Zhi-Wei Sun_, Aug 21 2013

%D Richard Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, Section F5.

%H Robin Chapman, <a href="http://dx.doi.org/10.4064/aa115-3-4">Determinants of Legendre symbol matrices</a>, Acta Arith. 115 (2004), 231-244.

%H Robin Chapman, <a href="http://dx.doi.org/10.1016/j.ejc.2003.02.001">Steinitz classes of unimodular lattices</a>, European J. Combin. 25 (2004), 487-493.

%H Robin Chapman (2009), <a href="http://secamlocal.ex.ac.uk/people/staff/rjchapma/etc/evildet.pdf">My evil determinant problem</a>

%H Maxim Vsemirnov (2011), <a href="http://arxiv.org/abs/1108.4031">On R. Chapman's ``evil determinant'': case p=1 (mod 4)</a>, arXiv:1108.4031 [math.NT], 2011-2012.

%H M. Vseminov, <a href="http://dx.doi.org/10.1016/j.laa.2011.08.039">On the evaluation of R. Chapman's "evil determinant"</a>, Linear Algebra Appl. 436(2012), 4101-4106.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Legendre_symbol">Legendre symbol</a>

%e p_4 = 7 = 2*3 + 1 and the 3 X 3 matrix (L((i+j)/7)) is

%e    1, -1,  1

%e   -1,  1, -1

%e    1, -1, -1

%e which has determinant 0, so a(4) = 0.

%t a[n_] := Module[{p, k}, p = Prime[n]; k = (p-1)/2; Det @ Table[JacobiSymbol[ i + j, p], {i, 1, k}, {j, 1, k}]];

%t Table[a[n], {n, 2, 32}] (* _Jean-François Alcover_, Nov 18 2018 *)

%Y Cf. A179071 (Chapman's "evil" determinants I), A179073 (A179071 for p == 1 (mod 4)), A179074 (A179072 for p == 1 (mod 4)).

%K sign

%O 2,2

%A _Jonathan Sondow_ and _Wadim Zudilin_, Jun 29 2010