Pieter Moree

Ternary cyclotomic polynomials having a large coefficient

Abstract Let Φ n (x) denote the nth cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that an (k), the coefficient of xk in Φ n (x), satisfies |an (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 |an (k)| ≦ 3p/4). Here we show that, nevertheless, Beiters conjecture is false for every p ≧ 11. We also prove that given any ε > 0 there exist infinitely many triples (pj , qj , rj ) with p 1 < p 2 < ⋯ consecutive primes such that |apjqjrj (nj )| > (2/3 – ε)pj for j ≧ 1.

Bertrand's Postulate for Primes in Arithmetical Progressions

Abstract Bertrands Postulate is the theorem that the interval ( x , 2 x ) contains at least one prime for x \2>1. We prove, building on work of Erd\”os, analogues of this result, in which the interval is of the form ( x, zx ) and there are at least m primes ≡ a (mod d ) required to be contained in this interval, and where z, a and d have to satisfy some conditions. For the case m  1 the results are worked out using a computer. They can be found in Table 1.

Cyclotomic numerical semigroups

Given a numerical semigroup

A Top Hat for Moser's Four Mathemagical Rabbits

S

Sequences of enumerative geometry: congruences and asymptotics

, we let

The lengths of Hermitian self-dual extended duadic codes

\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s

On the distribution of the order and index of g(modp) over residue classes II

be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups

Numerical Semigroups, Cyclotomic Polynomials, and Bernoulli Numbers

S

Geometry and Dynamics of Groups and Spaces

such that

Note: Values of coefficients of cyclotomic polynomials II

\mathrm P_S(x)