Related papers: Notes On a Borwein and Choi's conjecture of cyclot…

This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…

We prove that if $P(X) \in \mathbb{Z}[X]$ is an integer polynomial of degree $n$ and having $P(0) = 1$, then either $P(X)$ is a product of cyclotomic polynomials, or else at least one of the complex roots of $P$ belongs to the disk $|z|…

We say that a cyclotomic polynomial \Phi_{n}(x) has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of \Phi_{n}(x) and M(p) be the maximum of A(pqr). In 1968, Sister…

We say that a cyclotomic polynomial Phi_{n}(x) has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Phi_{n}(x) and M(p) be the maximum of A(pqr). In 1968, Sister…

This note describes a conjecture involving cyclotomic polynomials and some initial thoughts towards a solution. Given positive integers $m,n$, the conjecture is that either $\Phi_m(q)\leqslant\Phi_n(q)$ or $\Phi_m(q)\geqslant\Phi_n(q)$…

The largest coefficient (in absolute value) of a cyclotomic polynomial $\Phi_n$ is called its height $A(n)$. In case $p$ is a fixed prime it turns out that as $q$ and $r$ range over all primes satisfying $p<q<r$, the height $A(pqr)$ assumes…

Let $\Phi_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*}…

A unitary cyclotomic polynomial of order three is a polynomial of the form \[ \Phi^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$,…

Let $\Psi_n(x)$ be the monic polynomial having precisely all non-primitive $n$th roots of unity as its simple zeros. One has $\Psi_n(x)=(x^n-1)/\Phi_n(x)$, with $\Phi_n(x)$ the $n$th cyclotomic polynomial. The coefficients of $\Psi_n(x)$…

A Newman polynomial has all the coefficients in $\{ 0,1\}$ and constant term 1, whereas a Littlewood polynomial has all coefficients in $\{-1,1\}$. We call $P(X)\in\mathbb{Z}[X]$ a Borwein polynomial if all its coefficients belong to $\{…

In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ \sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2\frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}\equiv…

Let $p$ be a prime number. As a standard application of the irreducibility criterion of Eisenstein, it is well known that the $p$-th cyclotomic polynomial $\Phi_p(t)=1+t+\dots+t^{p-1}$ is the minimal polynomial of $e^{2\pi i/p}$ over…

We prove a recent conjecture due to Deutsch, Sagan, and Wilson stating that the finite sequence obtained from the first p central trinomial coefficients modulo p by replacing nonzero terms by 1's is palindromic, for any prime number p > 3.…

Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period $p^r$ was proposed by Z. Xiao et al. (Des. Codes Cryptogr., DOI 10.1007/s10623-017-0408-7). Later, for the case $f$ being…

We build a new theory for analyzing the coefficients of any cyclotomic polynomial by considering it as a gcd of simpler polynomials. Using this theory, we generalize a result known as periodicity and provide two new families of flat…

Let $\Phi_n^{(k)}(x)$ be the $k$-th derivative of $n$-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: $2\Phi^{(3)}_n(1)$ is divisible by $\phi(n)-2$ and $\Phi^{(2k+1)}_n(1)$ is divisible by…

Let $\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\Phi_n(x)$, satisfies $|a_n(k)|\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case…

We prove that for every $\varepsilon>0$ and a nonnegative integer $\omega$ there exist primes $p_1,p_2,\ldots,p_\omega$ such that for $n=p_1p_2\ldots p_\omega$ the height of the cyclotomic polynomial $\Phi_n$ is at least…

Recent work of Borwein, Choi, and the second author examined a collection of polynomials closely related to the Goldbach conjecture: the polynomial $F_N$ is divisible by the $N$th cyclotomic polynomial if and only if there is no…

It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to…