

A225611


Determinant of the (p_n1) X (p_n1) matrix with (i,j)entry equal to the Legendre symbol ((i^2+6*i*j+j^2)/p_n), where p_n is the nth prime.


2



0, 16, 0, 0, 0, 18939904, 0, 0, 600706205614080, 0, 3126394312091238400, 915844279166632469526048000, 0, 0, 1513783909437524991467008819200000000, 0, 6597762875255062617688526826958066024448000, 0, 0
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OFFSET

2,2


COMMENTS

Conjecture: We have a(n) = 0 if p_n == 3 (mod 4).
For an odd prime p and integers c and d, ZhiWei Sun defined (c,d)_p to be the determinant of the (p1) X (p1) matrix whose (i,j)entry is the Legendre symbol ((i^2+c*i*j+d*j^2)/p). It is easy to see that (c,d)_p = (1/p)*(c,d)_p. Sun conjectured that for any integer c and nonzero integer d there are infinitely many odd primes p with (c,d)_p = 0, moreover (c,d)_p = 0 if (d/p) = 1. He also formulated some concrete conjectures in the case (d/p) = 1. For example, (3,3)_p = 0 if p == 11 (mod 12), and (10,9)_p = 0 if p == 5 (mod 12); (3,2)_p = (4,2)_p = 0 if p == 7 (mod 8).
On August 12 2013, ZhiWei Sun conjectured that for any odd prime p and integers c and d with d not divisible by p, if (c,d)_p is nonzero then its padic valuation (i.e., padic order) must be even.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 2..50
ZhiWei Sun, On some determinants with Legendre symbol entries, preprint, August 2013.


MATHEMATICA

a[n_]:=Det[Table[JacobiSymbol[i^2+6*i*j+j^2, Prime[n]], {i, 1, (Prime[n]1)}, {j, 1, (Prime[n]1)}]]
Table[a[n], {n, 2, 20}]


CROSSREFS

Cf. A226163, A227609, A227968, A227971, A228005, A228077, A228095.
Sequence in context: A023919 A306282 A169767 * A173293 A008433 A010111
Adjacent sequences: A225608 A225609 A225610 * A225612 A225613 A225614


KEYWORD

sign


AUTHOR

ZhiWei Sun, Aug 11 2013


STATUS

approved



