A146961 Numbers k = p*q*r, with odd primes p < q < r, such that Sister Beiter's cyclotomic coefficient conjecture is false.

20213, 125609, 136477, 141317, 150271, 198493, 199177, 212971, 239039, 273229, 282367, 291343, 311201, 332777, 373901, 393313, 398563, 412357, 442091, 449527, 449647, 450131, 456569, 461263, 469249, 470741, 475057, 522461, 524837, 532363

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

Subsequence of A046389.

Sequence in context: A295014 A133527 A295011 * A295013 A295000 A250451

Adjacent sequences: A146958 A146959 A146960 * A146962 A146963 A146964

Keyword

nonn

Author

T. D. Noe, Nov 03 2008

Status

approved