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A179072 Chapman's "evil" determinants II. 6
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

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

REFERENCES

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

LINKS

Table of n, a(n) for n=2..32.

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 (2009), My evil determinant problem

Maxim Vsemirnov (2011), On R. Chapman's ``evil determinant'': case p=1 (mod 4), arXiv:1108.4031 [math.NT], 2011-2012.

M. Vseminov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436(2012), 4101-4106.

Wikipedia, Legendre symbol

EXAMPLE

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.

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-Fran├žois Alcover, Nov 18 2018 *)

CROSSREFS

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

Sequence in context: A287506 A288125 A020916 * A073111 A229685 A230469

Adjacent sequences:  A179069 A179070 A179071 * A179073 A179074 A179075

KEYWORD

sign

AUTHOR

Jonathan Sondow and Wadim Zudilin, Jun 29 2010

STATUS

approved

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Last modified November 14 14:43 EST 2019. Contains 329126 sequences. (Running on oeis4.)