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Acta Mathematica Sinica, English Series

, Volume 25, Issue 1, pp 65–76 | Cite as

Notes on the Borwein-Choi conjecture of Littlewood cyclotomic polynomials

  • Shao Fang HongEmail author
  • Wei Cao
Article

Abstract

Borwein and Choi conjectured that a polynomial P(x) with coefficients ±1 of degree N − 1 is cyclotomic iff
$$ P(x) = \pm \Phi _{p_1 } ( \pm x)\Phi _{p_2 } ( \pm x^{p_1 } ) \cdots \Phi _{p_r } ( \pm x^{p_1 p_2 \cdots p_{r - 1} } ), $$
, where N = p 1 p 2p r and the p i are primes, not necessarily distinct. Here Φ p (x):= (x p − 1)/(x − 1) is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the conjecture.

Keywords

cyclotomic polynomial Littlewood polynomial E-transformation Ramanujan sum least element 

MR(2000) Subject Classification

11R09 11Y99 

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References

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    Borwein, P., Choi, S., Ferguson, R.: Norms of cyclotomic Littlewood polynomials. Math. Proc. Camb. Phil. Soc., 138, 315–326 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Mathematical CollegeSichuan UniversityChengduP. R. China

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