A polynomial given by

(1) |

(2) |

(3) |

(4) |

(5) |

- 1. Iff for some and ,
- 2. Iff for and ,
- 3. otherwise .

The Logarithm of the cyclotomic polynomial

(6) |

The first few cyclotomic Polynomials are

The smallest values of for which has one or more coefficients , , , ... are 0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, ... (Sloane's A013594).

The Polynomial can be factored as

(7) |

(8) |

(9) |

can also be computed from

(10) |

**References**

Beiter, M. ``The Midterm Coefficient of the Cyclotomic Polynomial .'' *Amer. Math. Monthly* **71**, 769-770, 1964.

Beiter, M. ``Magnitude of the Coefficients of the Cyclotomic Polynomial .'' *Amer. Math. Monthly* **75**, 370-372, 1968.

Bloom, D. M. ``On the Coefficients of the Cyclotomic Polynomials.'' *Amer. Math. Monthly* **75**, 372-377, 1968.

Carlitz, L. ``The Number of Terms in the Cyclotomic Polynomial .'' *Amer. Math. Monthly* **73**, 979-981, 1966.

Conway, J. H. and Guy, R. K. *The Book of Numbers.* New York: Springer-Verlag, 1996.

de Bruijn, N. G. ``On the Factorization of Cyclic Groups.'' *Indag. Math.* **15**, 370-377, 1953.

Lam, T. Y. and Leung, K. H. ``On the Cyclotomic Polynomial .'' *Amer. Math. Monthly* **103**, 562-564, 1996.

Lehmer, E. ``On the Magnitude of Coefficients of the Cyclotomic Polynomials.'' *Bull. Amer. Math. Soc.* **42**, 389-392, 1936.

McClellan, J. H. and Rader, C. *Number Theory in Digital Signal Processing.* Englewood Cliffs, NJ: Prentice-Hall, 1979.

Migotti, A. ``Zur Theorie der Kreisteilungsgleichung.''
*Sitzber. Math.-Naturwiss. Classe der Kaiser. Akad. der Wiss., Wien* **87**, 7-14, 1883.

Schroeder, M. R.
*Number Theory in Science and Communication, with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed.*
New York: Springer-Verlag, p. 245, 1997.

Sloane, N. J. A. Sequence A013594 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Vardi, I. *Computational Recreations in Mathematica.* Redwood City, CA: Addison-Wesley, pp. 8 and 224-225, 1991.

© 1996-9

1999-05-25