

A179071


Chapman's "evil" determinants I.


6



1, 2, 1, 1, 18, 4, 1, 1, 70, 1, 882, 32, 1, 1, 182, 1, 29718, 1, 1, 1068, 1, 1, 500, 5604, 4030, 1, 1, 8890182, 776, 1, 1, 1744, 1, 113582, 1, 4832118, 1, 1, 1118, 1, 1111225770, 1, 1764132, 11018, 1, 1, 1, 1, 20000849130, 23156, 1
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OFFSET

2,2


COMMENTS

Determinant of the (k+1)X(k+1) matrix with (i,j)entry L((ji)/p), where L(./p) denotes the Legendre symbol modulo p and p = p_n = 2k+1 is the nth prime.
Guy says "Chapman has a number of conjectures which concern the distribution of quadratic residues." One is that if p_n == 3 (mod 4), then a(n) = 1. Chapman also has a conjecture if p_n == 1 (mod 4), involving the fundamental unit and class number of the quadratic field Q(sqrt(p)). (Added Aug 23 2011: Both conjectures have been proved by Vsemirnov.)
It appears that a(n) is negative and even, if p_n == 1 (mod 4); see A179073. (Added Aug 28 2011: This conjecture has also been proved by Vsemirnov.)


REFERENCES

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


LINKS

Table of n, a(n) for n=2..52.
Robin Chapman, Determinants of Legendre symbol matrices, Acta Arith. 115 (2004), 231244.
Robin Chapman, Steinitz classes of unimodular lattices, European J. Combin. 25 (2004), 487493.
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], 20112012.
M. Vseminov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436(2012), 41014106.
Wikipedia, Legendre symbol


EXAMPLE

p_3 = 5 = 2*2+1 and the (2+1)by(2+1) matrix (L((ji)/5)) is
0, 1, 1
1, 0, 1
1, 1, 0
which has determinant 2, so a(3) = 2.


MATHEMATICA

a[n_] := Module[{p, k}, p = Prime[n]; k = (p1)/2; Det @ Table[JacobiSymbol[ ji, p], {i, 1, k+1}, {j, 1, k+1}]];
Table[a[n], {n, 2, 52}] (* JeanFrançois Alcover, Nov 18 2018 *)


PROG

(PARI) a(n) = my(p=prime(n), k=(p+1)/2); matdet(matrix(k, k, i, j, kronecker(ji, p))); \\ Michel Marcus, Aug 25 2021


CROSSREFS

Cf. A179072 (Chapman's "evil" determinants II), A179073 (A179071 for p == 1 (mod 4)), A179074 (A179072 for p == 1 (mod 4)).
Sequence in context: A174918 A154991 A090163 * A124001 A174966 A157453
Adjacent sequences: A179068 A179069 A179070 * A179072 A179073 A179074


KEYWORD

sign


AUTHOR

Jonathan Sondow and Wadim Zudilin, Jun 29 2010


STATUS

approved



