Maximum gap in (inverse) cyclotomic polynomial

Hoon Hong, Eunjeong Lee, Hyang Sook Lee, Cheol Min Park

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12 Scopus citations


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 languageEnglish
Pages (from-to)2297-2315
Number of pages19
JournalJournal of Number Theory
Issue number10
StatePublished - 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.


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


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