CYCLOTOMIC POLYNOMIALS WITH PRESCRIBED HEIGHT AND PRIME NUMBER THEORY

On Prime Values of Cyclotomic Polynomials

We present several approaches on finding necessary and sufficient conditions on n so that Φk(x ) is irreducible where Φk is the k-th cyclotomic polynomial.

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Cyclotomic polynomials with large coefficients

The coefficients a(m,n) and especially A(n) and S(n) have been the subject of numerous investigations (see [1] and the references given there). Until recently all these investigations concerned very thin sets of integers n. In [3] the author could establish a property valid for a set of integers of asymptotic density 1. Let ε(n) be any function defined for all positive integers such that limn→∞...

Cyclotomic Polynomials

Some compact generalization of inequalities for polynomials with prescribed zeros

‎Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$‎, ‎$k^2 leq rRleq R^2$ and for $Rleq r leq k$‎. ‎Our results refine and generalize certain well-known polynomial inequalities‎.