cp's OEIS Frontend

A372409 Determinant of the matrix obtained from [Jacobi(i-j, 2*n+1)]_{0 by replacing all the entries in the first row by 1, where Jacobi(a,m) denotes the Jacobi symbol (a/m).

%I A372409 #23 May 01 2024 13:42:59
%S A372409 1,-1,1,-3,3,-8,-5,-5,-24,7,0,0,9,9,-81312,-1341867,11,-19685120,-13,
%T A372409 -13,0,-15,0,-180287762432,17,-1407939911477,10526233598464,19,19,0,
%U A372409 6040299856799,-21,29830847001120768,23,-23,0,115407361849089836,-25,0,27,104060523591574200
%N A372409 Determinant of the matrix obtained from [Jacobi(i-j, 2*n+1)]_{0<i,j<n} by replacing all the entries in the first row by 1, where Jacobi(a,m) denotes the Jacobi symbol (a/m).
%C A372409 Conjecture 1: If 2*n + 1 (with n > 1) is a prime p, then a(n)/floor((p-2)/3) coincides with (-1)^((p+3)/4) if p == 1 (mod 4), and (-1)^((h(-p)+1)/2) if p == 3 (mod 4), where h(-p) is the class number of the imaginary quadratic field Q(sqrt(-p)).
%C A372409 Conjecture 2: Let p > 3 be a prime, and let S(p) and T(p) denote the matrices obtained from [Jacobi(i+j,p)]_{1<=i,j<=(p-3)/2} and [Jacobi(i+j,p)]_{0<=i,j<=(p-3)/2} (respectively) by replacing all the entries in the first row by 1. Then det S(p) = -det T(p) = 2^((p-5)/2)*s(p), where s(p) is (-1)^((p+3)/4) if p == 1 (mod 4), and (-1)^((h(-p)-1)/2) if p == 3 (mod 4).
%C A372409 Both conjectures are motivated by Conjecture 4.6 in the author's 2019 FFA paper as well as the conjectures in A372385. They have been verified for primes p < 2000.
%D A372409 L.-Y. Wang and H.-L. Wu, On certain determinants involving Legendre symbols, Ramanujan J. 58 (2022), 43-56.
%H A372409 R. Chapman, <a href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_4064-aa115-3-4">Determinants of Legendre symbol matrices</a>, Acta Arith. 115 (2004), 231-244.
%H A372409 Z.-W. Sun, <a href="https://doi.org/10.1016/j.ffa.2018.12.004">On some determinants with Legendre symbol entries</a>, Finite Fields Appl. 56 (2019), 285-307.
%H A372409 M. Vsemirnov, <a href="https://doi.org/10.1016/j.laa.2011.08.039">On the evaluation of R. Chapman's "evil determinant"</a>, Linear Algebra Appl. 436 (2012), 4101-4106.
%H A372409 M. Vsemirnov, <a href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-4-3">On R. Chapman's "evil determinant": case p == 1(mod 4)</a>, Acta Arith. 159 (2013), 331-344; see also the <a href="https://arxiv.org/abs/1108.4031">arXiv version</a>, arXiv:1108.4031 [math.NT], 2011-2012.
%H A372409 L.-Y. Wang and H.-L. Wu, <a href="https://arxiv.org/abs/2401.05853">On p-th cyclotomic field and Zhi-Wei Sun's Legendre determinants over F_p</a>, arXiv:2401.05853 [math.NT], 2024.
%e A372409 a(3) = -1 since the determinant of the matrix [1, 1; Jacobi(2-1,2*3+1), Jacobi(2-2,2*3+1)] = [1, 1; 1, 0] has the value -1.
%t A372409 a[n_]:=a[n]=Det[Table[If[i==1,1,JacobiSymbol[i-j,2*n+1]],{i,1,n-1},{j,1,n-1}]];
%t A372409 tab={};Do[tab=Append[tab,a[n]],{n,2,42}];Print[tab]
%o A372409 (PARI) a(n) = matdet(matrix(n-1, n-1, i, j, if (i==1, 1, kronecker(i-j, 2*n+1)))); \\ _Michel Marcus_, Apr 30 2024
%o A372409 (Python)
%o A372409 from sympy import Matrix, jacobi_symbol
%o A372409 def A372409(n): return Matrix(n-1,n-1,[jacobi_symbol(i-j,(n<<1)|1) if i else 1 for i in range(n-1) for j in range(n-1)]).det() # _Chai Wah Wu_, May 01 2024
%Y A372409 Cf. A179071, A179072, A372385.
%K A372409 sign
%O A372409 2,4
%A A372409 _Zhi-Wei Sun_, Apr 29 2024