## Abstract

Let g(f) denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial f. Let Φ _{n} denote the n-th cyclotomic polynomial and let Ψ _{n} denote the n-th inverse cyclotomic polynomial. In this note, we study g(Φ _{n}) and g(Ψ _{n}) where n is a product of odd primes, say p _{1}<p _{2}<p _{3}, etc. It is trivial to determine g(Φp1), g(Ψp1) and g(Ψp1p2). Hence the simplest non-trivial cases are g(Φ _{p1p2}) and g(Ψ _{p1p2p3}). We provide an exact expression for g(Φ _{p1p2}). We also provide an exact expression for g(Ψ _{p1p2p3}) under a mild condition. The condition is almost always satisfied (only finite exceptions for each p _{1}). We also provide a lower bound and an upper bound for g(Ψ _{p1p2p3}).

Original language | English |
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Pages (from-to) | 2297-2315 |

Number of pages | 19 |

Journal | Journal of Number Theory |

Volume | 132 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2012 |

### Bibliographical note

Funding Information:Eunjeong Lee was supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2010-0028298). Hyang-Sook Lee and Cheol-Min Park were supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science and Technology. (No. 2010-0000402). We would like to thank the anonymous referees for their insightful and helpful suggestions.

## Keywords

- Cyclotomic polynomial
- Inverse cyclotomic polynomial
- Pairing-based cryptosystem