A174022 - OEIS

%I #10 Nov 17 2021 21:07:13

%S 43,67,163,173,293,331,379,463,487,499,547,643,677,683,773,797,823,

%T 853,883,907,941,947,967,1013,1051,1087,1097,1123,1163,1217,1229,1303,

%U 1423,1493,1523,1553,1567,1613,1637,1693,1723,1747,1787,1867,1877,1987,1997

%N Primes p for which the Fekete polynomial fp(x) has a zero between 0 and 1.

%C The Fekete polynomial fp(x) is defined as sum_{k=0..p-1} (k|p) x^k, where (k|p) is the Legendre symbol. Conrey et al. mention that there are 23 such primes less than 1000, which is verified here. The coefficients of the polynomial are in the rows of sequence A097343. It appears that zeros in (0,1) always come in pairs. As noted by Franz Lemmermeyer in Math Overflow, it appears that after factoring x, x-1, and x+1 out of fp(x), we are left with an irreducible polynomial.

%D Peter Borwein, Computational excursions in analysis and number theory, Springer-Verlag, 2002, Chap.5.

%H Charles R Greathouse IV, <a href="/A174022/b174022.txt">Table of n, a(n) for n = 1..246</a>

%H J. Brian Conrey, Andrew Granville, Bjorn Poonen, K. Soundararajan, <a href="http://arxiv.org/abs/math/9906214v1">Zeros of Fekete polynomials</a>, arXiv:math/9906214 [math.NT], 1999.

%H Math Overflow, <a href="http://mathoverflow.net/questions/17638/irreducibility-of-polynomials-related-to-quadratic-residues">Irreducibility of polynomials related to quadratic residues</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fekete_polynomial">Fekete polynomial</a>

%t t={}; Do[poly=JacobiSymbol[Range[0,p-1], p].x^Range[0,p-1]; FactorOut[0]; FactorOut[1]; FactorOut[1]; FactorOut[ -1]; c=CountRoots[poly,{x,0,1}]; If[c>0, AppendTo[t,p]], {p,Prime[Range[PrimePi[1000]]]}]; t

%o (PARI) Fekete(p)=Pol(vector(p,a,kronecker(a,p)))

%o is(p)=my(x='x,P=Fekete(p)/x); P/=(x-1)^valuation(P,x-1); polsturm(P,[0,1])>0 \\ _Charles R Greathouse IV_, Nov 12 2021

%K nonn

%O 1,1

%A _T. D. Noe_, Mar 11 2010