The family of ternary cyclotomic polynomials with one free prime

Abstract

A cyclotomic polynomial n (x) is said to be ternary if n = pqr , with p, q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behavior of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family pqr (x) with p < q < r , p and q fixed and r a free prime.

The Height of a Class of Ternary Cyclotomic Polynomials

“…Considering Möller's result with p = 5 and using the general fact A(5qr) ≤ 3 (established independently by Beiter [3] and Bloom [5]), we obtain that A(5qr) = 3 when q ≡ 2 (mod 5) and 2r ≡ −1 (mod 5q). We refer the reader to the paper of Gallot, Moree and Wilms [9] which gives a more detailed description of A(5qr).…”

“…(3) In 2011, Gallot, Moree and Wilms [9] proved that if p ≥ 5 and 2p − 1 is a prime, then for appropriate r, A(p(2p − 1)r) = 3.…”

Sister Beiter and Kloosterman: A tale of cyclotomic coefficients and modular inverses

“…We remark that a constant c for which the left hand side of (14) is less than c p 3/4 log p must be larger than 2 √ 2, but for sufficiently large p, it can be chosen as close to 2 √ 2 = 2.828427125... as one wishes. Numerical computations for smaller p show that if c = 2.8320056 the estimations hold for all prime numbers p. This completes the proof of the lemma.…”

“…We say that a function is ultimately constant on an infinite sequences of integers, if it takes on the same value for all sufficiently large elements in the sequence. The study of M (p; q) was initiated by Gallot et al [14]. The main conjecture is that given a prime p, there exists a modulus d p , such that M (p; q) is ultimately constant on every primitive residue class modulo d p .…”

Cyclotomic coefficients: gaps and jumps

“…Recently, there has been a burst of activity in studying the cyclotomic coefficients a n (k), see, for example, [3,4,5,6,10,15,17,18,19,31,35] and references therein. Furthermore, in several works inverse cyclotomic polynomials Ψ n (Z) = (Z n − 1)/Φ n (Z) have also been considered, see [7,21,22,23,29].…”