Neighboring ternary cyclotomic coefficients differ by at most one

2011

A cyclotomic polynomial Φ 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 behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ pqr (x) with p < q < r, p and q fixed and r a free prime.

2008

Let Φ n (x) denote the nth cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that a n (k), the coefficient of x k in Φ n (x), satisfies |a n (k)| ≤ (p + 1)/2 in case n = pqr with p < q < r primes (in this case Φ n (x) is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example |a n (k)| ≤ 3p/4). Here we show that, nevertheless, Beiter's conjecture is false for every p ≥ 11. We also prove that given any ǫ > 0 there exist infinitely many triples (p j , q j , r j ) with p 1 < p 2 < . . . consecutive primes such that |a p j q j r j (n j )| > (2/3ǫ)p j for j ≥ 1.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2009

Let Φ n (x) denote the nth cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that a n (k), the coefficient of x k in Φ n (x), satisfies |a n (k)| ≤ (p + 1)/2 in case n = pqr with p < q < r primes (in this case Φ n (x) is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example |a n (k)| ≤ 3p/4). Here we show that, nevertheless, Beiter's conjecture is false for every p ≥ 11. We also prove that given any ǫ > 0 there exist infinitely many triples (p j , q j , r j ) with p 1 < p 2 < . . . consecutive primes such that |a p j q j r j (n j )| > (2/3ǫ)p j for j ≥ 1.

Involve, a Journal of Mathematics, 2011

A cyclotomic polynomial Φ 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 behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ pqr (x) with p < q < r, p and q fixed and r a free prime.

2019

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials Φ n pxq. They can be written as certain products of cyclotomic poynomials. We study the case where n has two or three distinct prime factors using numerical semigroups, respectively Bachman’s inclusion-exclusion polynomials. Given m ě 1 we show that every integer occurs as a coefficient of Φ mn pxq for some n ě 1 following Ji, Li and Moree [9]. Here n will typically have many different prime factors. We also consider similar questions for the polynomials pxn ́ 1q{Φ n pxq, the inverse unitary cyclotomic polynomials.

International Mathematical Forum, 2011

Glasgow Mathematical Journal, 1985

We define the nth cyclotomic polynomial Φn(z) by the equationand we writewhere ϕ is Euler&#39;s function.Erdös and Vaughan [3] have shown thatuniformly in n as m-→∞, whereand that for every large m

Journal of Number Theory, 2016

We improve several recent results by Hong, Lee, Lee and Park (2012) on gaps and Bzdȩga (2014) on jumps amongst the coefficients of cyclotomic polynomials. Besides direct improvements, we also introduce several new techniques that have never been used in this area. 2010 Mathematics Subject Classification. 11B83, 11L07, 11N25. Key words and phrases. Coefficients of cyclotomic polynomials, products of primes, numerical semigroups, double Kloosterman sums.

The Electronic Journal of Combinatorics, 2012

We pose the question of determining the lowest-degree polynomial with nonnegative coefficients divisible by the $n$-th cyclotomic polynomial $\Phi_n(x)$. We show this polynomial is $1 + x^{n/p} + \cdots + x^{(p-1)n/p}$ where $p$ is the smallest prime dividing $n$ whenever $2/p &gt; 1/q_1 + \cdots + 1/q_k$, where $q_1, \ldots, q_k$ are the other (distinct) primes besides $p$ dividing $n$. Determining the lowest-degree polynomial with nonnegative coefficients divisible by $\Phi_n(x)$ remains open in the general case, though we conjecture the existence of values of $n$ for which this degree is, in fact, less than $(p-1)n/p$.

Mathematika, 2021

Given any positive integer n, let A(n) denote the height of the n th cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that A(n) is unbounded. We conjecture that every natural number can arise as value of A(n) and prove this assuming that for every pair of consecutive primes p and p ′ with p ≥ 127 we have p ′ −p < √ p+1. We also conjecture that every natural number occurs as the maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture holds, i.e., that √ p ′ − √ p < 1 always holds. This is the first time, as far as the authors know, that a connection between prime gaps and cyclotomic polynomials is uncovered. Using a result of Heath-Brown on prime gaps we show unconditionally that every natural number m ≤ x occurs as A(n) value with at most O ǫ (x 3/5+ǫ) exceptions. On the Lindelöf Hypothesis we show there are at most O ǫ (x 1/2+ǫ) exceptions and study them further by using deep work of Bombieri-Friedlander-Iwaniec on the distribution of primes in arithmetic progressions beyond the square-root barrier.