1. INTRODUCTION. In the seventeenth century, Fermat defined the sequence of numbers F n = 2 2 n + 1 for n â‰¥ 0, now known as Fermat numbers. If F n happens to be prime, F n is called a Fermat prime.â€¦ (More)

Let G be a finite group and let x âˆˆ G. Define the order subset of G determined by x to be the set of all elements in G that have the same order as x. A group G is said to have perfect order subsetsâ€¦ (More)

Using an idea of ErdÅ‘s, SierpiÅ„ski proved that there exist infinitely many odd positive integers k such that k Â· 2 + 1 is composite for all positive integers n. In this paper we give a briefâ€¦ (More)

We describe a recursive algorithm that produces an integral basis for the centre of the Hecke algebra of type A consisting of linear combinations of monomial symmetric polynomials of Jucysâ€“Murphyâ€¦ (More)

In 1960, SierpiÅ„ski proved that there exist infinitely many odd positive rational integers k such that k Â· 2n + 1 is composite in Z for all n â‰¥ 1. Any such integer k is now known as a SierpiÅ„skiâ€¦ (More)

A classical theorem in number theory due to Euler states that a positive integer z can be written as the sum of two squares if and only if all prime factors q of z, with q â‰¡ 3 (mod 4), occur withâ€¦ (More)

We develop basic properties of groups that contain exactly k elements of order n and are generated by these elements (called (k, n)-homogeneous groups). We then give a complete classification of (n,â€¦ (More)