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A179072 Chapman's "evil" determinants II.

%I A179072 #32 May 24 2025 10:35:54
%S A179072 -1,-2,0,0,-32,256,0,0,-8192,0,-262144,5242880,0,0,-33554432,0,
%T A179072 -2684354560,0,0,8589934592000,0,0,932385860354048,160159261748363264,
%U A179072 -1125899906842624,0,0,-225179981368524800,5260204364768739328,0,0
%N A179072 Chapman's "evil" determinants II.
%C A179072 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 A179072 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 A179072 It appears that a(n) is even, if p_n == 1 (mod 4).
%C A179072 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 A179072 Richard Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, Section F5.
%H A179072 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 A179072 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 A179072 Robin Chapman, <a href="http://secamlocal.ex.ac.uk/people/staff/rjchapma/etc/evildet.pdf">My evil determinant problem</a>, 2009.
%H A179072 Maxim Vsemirnov, <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 A179072 Maxim Vsemirnov, <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 A179072 Wikipedia, <a href="http://en.wikipedia.org/wiki/Legendre_symbol">Legendre symbol</a>
%e A179072 p_4 = 7 = 2*3 + 1 and the 3 X 3 matrix (L((i+j)/7)) is
%e A179072    1, -1,  1
%e A179072   -1,  1, -1
%e A179072    1, -1, -1
%e A179072 which has determinant 0, so a(4) = 0.
%t A179072 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 A179072 Table[a[n], {n, 2, 32}] (* _Jean-François Alcover_, Nov 18 2018 *)
%Y A179072 Cf. A179071 (Chapman's "evil" determinants I), A179073 (A179071 for p == 1 (mod 4)), A179074 (A179072 for p == 1 (mod 4)).
%K A179072 sign
%O A179072 2,2
%A A179072 _Jonathan Sondow_ and _Wadim Zudilin_, Jun 29 2010