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A NOTE ON TERNARY CYCLOTOMIC POLYNOMIALS

  • Zhang, Bin (School of Mathematical Sciences Nanjing Normal University)
  • Received : 2013.06.25
  • Published : 2014.07.31

Abstract

Let ${\Phi}_n(x)={\sum}^{{\phi}(n)}_{k=0}a(n,k)x^k$ denote the n-th cyclotomic polynomial. In this note, let p < q < r be odd primes, where $q{\not{\equiv}}1$ (mod p) and $r{\equiv}-2$ (mod pq), we construct an explicit k such that a(pqr, k) = -2.

Keywords

References

  1. G. Bachman, Flat cyclotomic polynomials of order three, Bull. London Math. Soc. 38 (2006), no. 1, 53-60. https://doi.org/10.1112/S0024609305018096
  2. G. Bachman and P. Moree, On a class of ternary inclusion-exclusion polynomials, Integers 11 (2011), A8, 14 pp.
  3. B. Bzdega, On the height of cyclotomic polynomials, Acta Arith. 152 (2012), no. 4, 349-359. https://doi.org/10.4064/aa152-4-2
  4. L. Carlitz, The number of terms in the cyclotomic polynomial $F{pq}(X)$, Amer. Math. Monthly 73 (1966), 979-981. https://doi.org/10.2307/2314500
  5. S. Elder, Flat Cyclotomic Polynomials: A New Approach, arXiv:1207.5811v1, 2012.
  6. H. Hong, E. Lee, H. S. Lee, and C. M. Park, Maximum gap in (inverse) cyclotomic polynomial, J. Number Theory 132 (2012), no. 10, 2297-2315. https://doi.org/10.1016/j.jnt.2012.04.008
  7. N. Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), no. 1, 118-126. https://doi.org/10.1016/j.jnt.2007.01.008
  8. T. Y. Lam and K. H. Leung, On the cyclotomic polynomial ${\Phi}{pq}(X)$, Amer. Math. Monthly 103 (1996), no. 7, 562-564. https://doi.org/10.2307/2974668
  9. E. Lehmer, On the magnitude of the coefficients of the cyclotomic polynomials, Bull. Amer. Math. Soc. 42 (1936), no. 6, 389-392. https://doi.org/10.1090/S0002-9904-1936-06309-3
  10. H. W. Lenstra, Vanishing sums of roots of unity, in: Proceedings, Bicentennial Congress Wiskundig Genootschap (Vrije Univ., Amsterdam, 1978), Part II, pp. 249-268, Math. Centre Tracts, 101, Math. Centrum, Amsterdam, 1979.
  11. H. Moller, Uber die Koeffizienten des n-ten Kreisteilungspolynoms, Math. Z. 119 (1971), 33-40. https://doi.org/10.1007/BF01110941
  12. P. Moree, Inverse cyclotomic polynomials, J. Number Theory 129 (2009), no. 3, 667-680. https://doi.org/10.1016/j.jnt.2008.10.004
  13. R. Thangadurai, On the coefficients of cyclotomic polynomials, in: Cyclotomic fields and related topics (Pune, 1999), 311-322, Bhaskaracharya Pratishthana, Pune, 2000.
  14. J. Zhao and X. K. Zhang, Coefficients of ternary cyclotomic polynomials, J. Number Theory 130 (2010), no. 10, 2223-2237. https://doi.org/10.1016/j.jnt.2010.03.012

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  1. ON A CLASS OF TERNARY CYCLOTOMIC POLYNOMIALS vol.52, pp.6, 2015, https://doi.org/10.4134/BKMS.2015.52.6.1911