A179072 Chapman's "evil" determinants II.

-1, -2, 0, 0, -32, 256, 0, 0, -8192, 0, -262144, 5242880, 0, 0, -33554432, 0, -2684354560, 0, 0, 8589934592000, 0, 0, 932385860354048, 160159261748363264, -1125899906842624, 0, 0, -225179981368524800, 5260204364768739328, 0, 0

Offset: 2

Author

Jonathan Sondow and Wadim Zudilin, Jun 29 2010

Keywords

• sign

Comments

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.

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.

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

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

Examples

			p_4 = 7 = 2*3 + 1 and the 3 X 3 matrix (L((i+j)/7)) is
   1, -1,  1
  -1,  1, -1
   1, -1, -1
which has determinant 0, so a(4) = 0.
		

References

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

Links

• Robin Chapman, Determinants of Legendre symbol matrices, Acta Arith. 115 (2004), 231-244.
• Robin Chapman, Steinitz classes of unimodular lattices, European J. Combin. 25 (2004), 487-493.
• Robin Chapman, My evil determinant problem, 2009.
• Maxim Vsemirnov, On R. Chapman's "evil determinant": case p=1 (mod 4), arXiv:1108.4031 [math.NT], 2011-2012.
• Maxim Vsemirnov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436(2012), 4101-4106.
• Wikipedia, Legendre symbol

Crossrefs

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

Programs

Mathematica

a[n_] := Module[{p, k}, p = Prime[n]; k = (p-1)/2; Det @ Table[JacobiSymbol[ i + j, p], {i, 1, k}, {j, 1, k}]];
Table[a[n], {n, 2, 32}] (* Jean-François Alcover, Nov 18 2018 *)