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Flat cyclotomic polynomials of order four and higher. (English) Zbl 1200.11020

Let \(\Phi_n\) denote the \(n\)th cyclotomic polynomial and let \(a_n(k)\) be its \(k\)th coefficient. Let \(V_n\) denote the set of coefficients of \(\Phi_n\). The cyclotomic polynomial \(\Phi_n\) is said to be flat if \(v\in V_n\) implies \(|v|\leq 1\). If \(n\) has order at most two, then \(\Phi_n\) is known to be flat. (The order of \(\Phi_n\) is the number of odd primes dividing \(n\).) Earlier the author in [J. Number Theory 127, No. 1, 118–126 (2007; Zbl 1171.11015)] showed that if \(p<q<r\) are odd primes with \(r\equiv \pm 1(\text{mod~}pq)\), then \(\Phi_{pqr}\) is flat, giving rise to an infinite family of flat cyclotomic polynomials of order three (the first such family was provided by G. Bachman). Here the author provides the first infinite family of flat cyclotomic polynomials of order four. The proof rests on the following main result in the paper: if \(n\) is odd and squarefree, and \(t>s>n\) are primes satisfying \(s\equiv t\pmod n\), then \(V_{ns}=V_{nt}\).

MSC:

11B83 Special sequences and polynomials
11C08 Polynomials in number theory
11N56 Rate of growth of arithmetic functions

Citations:

Zbl 1171.11015
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