A specific family of cyclotomic polynomials of order three

Abstract

Let A(n) be the largest absolute value of any coefficient of n-th cyclotomic polynomial Φn(x). We say Φn(x) is flat if A(n) = 1. In this paper, for odd primes p < q < r and 2r ≡ ±1 (mod pq), we prove that Φpqr(x) is flat if and only if p = 3 and q ≡ 1 (mod 3).

The Height of a Class of Ternary Cyclotomic Polynomials

“…This result was generalized by Flanagan [8] and improved by Kaplan [11]. There have been also studies of Φ pqr (x) with A(pqr) = 1, see [6,7,10,16]. In [11], Kaplan established the following periodicity of the function A(pqr).…”

A Survey on Coefficients of Cyclotomic Polynomials

“…Kaplan (2007) [64, Lemma 1] proved the following lemma, which provides a formula for the coefficients of ternary cyclotomic polynomials. This is known as Kaplan's lemma and has been used to prove several results on ternary cyclotomic polynomials [37,48,49,51,60,89,100,101,102,105,106]. Lemma 3.1 (Kaplan's lemma).…”

“…Ji (2010) [60] considered odd primes p < q < r such that 2r ≡ ±1 (mod pq) and showed that in such a case Φ pqr (X) is flat if and only if p = 3 and q ≡ 1 (mod 3). For a ∈ {3, 4, 5}, Zhang (2017) [102] gave similar characterizations for the odd primes such that ar ≡ ±1 (mod pq) and Φ pqr (X) is flat (see also [106] for a weaker result for the case a = 4).…”

The family of ternary cyclotomic polynomials with one free prime

“…We say Φ n (x) is flat if A(n) = 1. ChunGang Ji [18] proved that if p < q < r are odd prime and 2r ≡ ±1(mod pq), then Φ pqr (x) is flat iff p = 3 and q ≡ 1(mod 3). It follows that M(p; q) ≥ 2 for p > 3.…”