Informatics Publishing Limited

Yves Gallot ¹, Pieter Moree ²

Affiliations
1 I2 Bis Rue Perrey, 31400 Toulouse, FR
2 Max-Planck-Institut Fur Mathematik, Vivatsgasse 7, D-53111 Bonn, DE

Abstract

A cyclotomic polynomial Φ_n(x) is said to be ternary if n=pqr with p, q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, the set of coefficients occurring in a ternary cyclotomic polynomial consists of consecutive integers.
As an application we reprove in a simpler way a result of Bachman from 2004 on ternary cyclotomic polynomials with an optimally large set of coefficients.

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Alexandru Ciolan ¹, Florian Luca ², Pieter Moree ³

Affiliations
1 Rheinische Friedrich-Wilhelms-Universitat Bonn, Regina Pacis Weg 3, D-53113 Bonn, DE
2 School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, ZA
3 Max Planck Institut fur Mathematik, Vivatsgasse 7, D-53111 Bonn, DE

Abstract

Given a recurrent sequence U := {U_n}_n≥0 we consider the problem of counting M_U(x), the number of integers n ≤ x such that U_n = u² + nv² for some integers u, v. We will show that MU(x) ⪻ x(log x)^−0.05 for a large class of ternary sequences. Our method uses many ingredients from the proof of Alba Gonzalez and the second author [1] that M_F (x) ⪻ x(log x)^−0.06, with F the Fibonacci sequence.

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