A Note on Ternary Cyclotomic Polynomials

Abstract

Abstract. Let Φn(x) = φ(n) k=0 a(n, k)x k denote the n-th cyclotomic polynomial. In this note, let p < q < r be odd primes, where q ≡ 1 (mod p) and r ≡ −2 (mod pq), we construct an explicit k such that a(pqr, k) = −2.

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).…”

“…[103] also showed that if p ≡ 1 (mod w), q ≡ 1 (mod pw), and r ≡ w (mod pq), for some integer w ≥ 2, then A + (pqr) = 1. Furthermore, for q ≡ 1 (mod p) and r ≡ −2 (mod pq), Zhang (2014) [100] constructed an explicit j such that a pqr (j) = −2, so that Φ pqr (X) is not flat. Regarding nonflat ternary cyclotomic polynomials with small heights, Zhang (2017) [101] showed that for every prime p ≡ 1 (mod 3) there exist infinitely many q and r such that A(pqr) = 3.…”

The Height of a Class of Ternary Cyclotomic Polynomials

“…Without fixing p, the first infinite family of ternary cyclotomic polynomials Φ pqr (x) with height exactly 2 was given by Elder [7], which showed that if q ≡ 1 (mod p) and r ≡ ±2 (mod pq), then A(pqr) = 2 (see Zhang [15] for another proof of this result).…”