A174022 Primes p for which the Fekete polynomial fp(x) has a zero between 0 and 1.

43, 67, 163, 173, 293, 331, 379, 463, 487, 499, 547, 643, 677, 683, 773, 797, 823, 853, 883, 907, 941, 947, 967, 1013, 1051, 1087, 1097, 1123, 1163, 1217, 1229, 1303, 1423, 1493, 1523, 1553, 1567, 1613, 1637, 1693, 1723, 1747, 1787, 1867, 1877, 1987, 1997

Offset

1,1

Comments

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.

References

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..246

J. Brian Conrey, Andrew Granville, Bjorn Poonen, K. Soundararajan, Zeros of Fekete polynomials, arXiv:math/9906214 [math.NT], 1999.

Math Overflow, Irreducibility of polynomials related to quadratic residues

Wikipedia, Fekete polynomial

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A043988 A139499 A201688 * A033229 A139875 A174812

Adjacent sequences: A174019 A174020 A174021 * A174023 A174024 A174025

Keyword

nonn

Author

T. D. Noe, Mar 11 2010

Status

approved