Cyclotomic numbers, coding theory and orthogonal polynomials

Numerical Semigroups, Cyclotomic Polynomials, and Bernoulli Numbers

We give two proofs of a folkore result relating numerical semigroups of embedding dimension two and binary cyclotomic polynomials and explore some consequences. In particular, we give a more conceptual reproof of a result of Hong et al. (2012) on gaps between the exponents of non-zero monomials in a binary cyclotomic polynomial. The intent of the author with this paper is to popularise cyclotom...

Fibonacci numbers and orthogonal polynomials

We prove that the sequence (1/Fn+2)n≥0 of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little q-Jacobi polynomials with q = (1− √ 5)/(1+ √ 5). We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices (1/F...

Quintic Polynomials and Real Cyclotomic Fields with Large Class Numbers

We study a family of quintic polynomials discoverd by Emma Lehmer. We show that the roots are fundamental units for the corresponding quintic fields. These fields have large class numbers and several examples are calculated. As a consequence, we show that for the prime p = 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401. In an a...

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