The family of ternary cyclotomic polynomials

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.

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.

A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, the set of coefficients occurring in a ternary cyclotomic polynomial consists of consecutive integers. As an application we reprove in a simpler way a result of Bachman from 2004 on ternary cyclotomic polynomials with an optimally large set of coefficients.

International Mathematical Forum, 2011

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.

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, 1982

We study properties of the polynomials Ok(X) which appear in the formal development 1 I;-o (a + bX')'" = xkiO tir(X) ar '6'. where rk E L' and r = 1 ri. This permits us to obtain the coefftcients of all cyclotomic polynomials. Then we use these properties to expand the cyclotomic numbers G,(i) = rIf : (a + bc")"'. where p is a prime, l is a primitive pth root of I. a, b E ,' and 1 <r <p-3. modulo powers of c-I (until (< ~ I)""-" r). This gives more information than the usual logarithmic derivative. Suppose that pkab(a + b). Let m =-b/a. We prove that G,(T) = cp modp(<-I)I for some CE P, if and only if \'[=I 1 kP-2 rmk E 0 (modp). We hope to show in this work that this result is useful in the study of the first case of Fermat's last theorem. Let p be a prime number, p > 5, and r be a primitive pth root of the unity. In this introduction we mention certain facts on Q(r) in connection with Fermat's Last Theorem. They led us to search for a more adequate expansion of the cyclotomic number j', (a+b{k)kr, a,bEJ, rEN. In Section I we give a generalization of Newton's Theorem on Binomial Expansion by studying the properties of the polynomials #Jx) such that /I; (a + bXk)Q = 1' qdk(X) a' kbh, ky0

Kodai Mathematical Journal, 2020

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials Φnpxq. 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 Φm n 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 px n´1 q{Φnpxq, the inverse unitary cyclotomic polynomials.

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 $(p-1)n/p$.