OFFSET
1,1
COMMENTS
In 1968, Sister Beiter conjectured that for k = p*q*r, with odd primes p < q < r, the maximum coefficient (in absolute value) of the cyclotomic polynomial Phi(k,x) is <= (p+1)/2. Up to 10^6, all counterexamples have p > 7. Gallot and Moree prove the conjecture is false for p > 7.
LINKS
Robin Visser, Table of n, a(n) for n = 1..200
A. S. Bang, Om Ligningen phi_n(x) = 0, Nyt tidsskrift for matematik, Vol. 6, Afdeling B (1895), pp. 6-12 (7 pages).
Yves Gallot and Pieter Moree, Counter-examples to Sister Beiter's cyclotomic coefficient conjecture, MPIM Preprint Series 2007 (141).
Nathan Kaplan, Flat cyclotomic polynomials of order three, Journal of Number Theory, Volume 127, Issue 1, November 2007, Pages 118-126.
G. S. Kazandzidis, On the cyclotomic polynomial: Coefficients, Bull. Soc. Math. Gr`ece (N.S.) 4 (1963), no. 1, 1-11.
Carlo Sanna, A Survey on Coefficients of Cyclotomic Polynomials, arXiv:2111.04034 [math.NT], 2021.
Wikipedia, Marion Beiter.
PROG
(PARI) isok(m) = if ((m%2) && (bigomega(m)==3) && (omega(m)==3), my(p=vecmin(factor(m)[, 1])); vecmax(apply(abs, Vec(polcyclo(m)))) > (p+1)/2; ); \\ Michel Marcus, Jan 16 2023
(Sage)
from sage.rings.polynomial.cyclotomic import cyclotomic_coeffs
for n in range(3, 100000, 2):
pqr = Integer(n).prime_factors()
if (len(pqr) == 3) and (product(pqr) == n):
coeffs = cyclotomic_coeffs(n, sparse=False)
max_coeff = max(abs(c) for c in coeffs)
if (max_coeff > (pqr[0]+1)//2): print(n) # Robin Visser, Aug 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Nov 03 2008
STATUS
approved