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

Author

T. D. Noe, Nov 03 2008

Keywords

• nonn

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.

Crossrefs

Subsequence of A046389.

Programs

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