Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON A CLASS OF TERNARY CYCLOTOMIC POLYNOMIALS

  • ZHANG, BIN (School of Mathematical Sciences Qufu Normal University) ;
  • ZHOU, YU (School of Mathematical Sciences Nanjing Normal University)
  • Received : 2014.07.08
  • Published : 2015.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

A cyclotomic polynomial ${\Phi}_n(x)$ is said to be ternary if n = pqr for three distinct odd primes p < q < r. Let A(n) be the largest absolute value of the coefficients of ${\Phi}_n(x)$. If A(n) = 1 we say that ${\Phi}_n(x)$ is flat. In this paper, we classify all flat ternary cyclotomic polynomials ${\Phi}_{pqr}(x)$ in the case $q{\equiv}{\pm}1$ (mod p) and $4r{\equiv}{\pm}1$ (mod pq).

Keywords

References

  1. A. Arnold and M. Monagan, Data on the heights and lengths of cyclotomic polynomials, Available: http://oldweb.cecm.sfu.ca/-ada26/cyclotomic/data.html.
  2. G. Bachman, Flat cyclotomic polynomials of order three, Bull. London Math. Soc. 38 (2006), no. 1, 53-60. https://doi.org/10.1112/S0024609305018096
  3. G. Bachman and P. Moree, On a class of ternary inclusion-exclusion polynomials, Integers 11 (2011), 1-14. https://doi.org/10.1515/integ.2011.001
  4. A. S. Bang, Om Lingingen ${\Phi}_n$(x) = 0, Tidsskr. Math. 6 (1895), 6-12.
  5. M. Beiter, Coefficients of the cyclotomic polynomial $F_{3qr}$(x), Fibonacci Quart. 16 (1978), no. 4, 302-306.
  6. D. M. Bloom, On the coefficients of the cyclotomic polynomials, Amer. Math. Monthly 75 (1968), 372-377. https://doi.org/10.2307/2313417
  7. D. Broadhurst, Flat ternary cyclotomic polynomials, Available: http://tech.groups. yahoo.com/group/primenumbers/message/20305.
  8. B. Bzdega, Jumps of ternary cyclotomic coefficients, Acta Arith. 163 (2014), no. 3, 203-213. https://doi.org/10.4064/aa163-3-2
  9. C. Cobeli, Y. Gallot, P. Moree, and A. Zaharescu, Sister Beiter and Kloosterman: A tale of cyclotomic coefficients and modular inverses, Indag. Math. (N.S.) 24 (2013), no. 4, 915-929. https://doi.org/10.1016/j.indag.2013.01.002
  10. D. Duda, The maximal coefficient of ternary cyclotomic polynomials with one free prime, Int. J. Number Theory 10 (2014), no. 4, 1067-1080. https://doi.org/10.1142/S1793042114500158
  11. S. Elder, Flat cyclotomic polynomials: A new approach, arXiv:1207.5811v1, 2012.
  12. T. Flanagan, On the coefficients of ternary cyclotomic polynomials, MS Thesis, University of Nevada Las Vegas, 2006.
  13. Y. Gallot and P. Moree, Ternary cyclotomic polynomials having a large coefficient, J. Reine Angew. Math. 632 (2009), 105-125.
  14. Y. Gallot, P. Moree, and R. Wilms, The family of ternary cyclotomic polynomials with one free prime, Involve 4 (2011), no. 4, 317-341. https://doi.org/10.2140/involve.2011.4.317
  15. 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
  16. C. G. Ji, A special family of cyclotomic polynomials of order three, Sci. China Math. 53 (2010), no. 9, 2269-2274. https://doi.org/10.1007/s11425-010-3148-y
  17. 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
  18. 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
  19. 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
  20. H. Moller, Uber die Koeffizienten des n-ten Kreisteilungspolynoms, Math. Z. 119 (1971), 33-40. https://doi.org/10.1007/BF01110941
  21. P. Moree and E. Rosu, Non-Beiter ternary cyclotomic polynomials with an optimally large set of coefficients, Int. J. Number Theory 8 (2012), no. 8, 1883-1902. https://doi.org/10.1142/S1793042112501072
  22. R. Thangadurai, On the coefficients of cyclotomic polynomials, In: Cyclotomic fields and related topics (Pune, 1999), 311-322, Bhaskaracharya Pratishthana, Pune, 2000.
  23. B. Zhang, A note on ternary cyclotomic polynomials, Bull. Korean Math. Soc. 51 (2014), no. 4, 949-955. https://doi.org/10.4134/BKMS.2014.51.4.949
  24. 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

Cited by

  1. Remarks on the flatness of ternary cyclotomic polynomials vol.13, pp.02, 2017, https://doi.org/10.1142/S1793042117501354