On the Littlewood cyclotomic polynomials

Shabnam Akhtari, Stephen K. Choi

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In this article, we study the cyclotomic polynomials of degree N - 1 with coefficients restricted to the set {+ 1, - 1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p (x) with coefficients ±1 of even degree N - 1 is cyclotomic if and only if p (x) = ± Φp1 (± x) Φp2 (± xp1) ⋯ Φpr (± xp1 p2 ⋯ pr - 1), where N = p1 p2 ⋯ pr and the pi are primes, not necessarily distinct. Here Φp (x) : = frac(xp - 1, x - 1) is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree 2α pβ - 1 with odd prime p or separable polynomials of any odd degree.

Original languageEnglish (US)
Pages (from-to)884-894
Number of pages11
JournalJournal of Number Theory
Issue number4
StatePublished - Apr 2008

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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