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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 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-François 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 December 1 07:59 EST 2021. Contains 349426 sequences. (Running on oeis4.)