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1. INTRODUCTION. In the seventeenth century, Fermat defined the sequence of numbers F n = 2 2 n + 1 for n ≥ 0, now known as Fermat numbers. If F n happens to be prime, F n is called a Fermat prime. Fermat showed that F n is prime for each n ≤ 4, and he conjectured that F n is prime for all n (see Brown [1] or Burton [2, p. 271]). Almost one hundred years(More)
Let D be a list of single digits, and let k be a positive integer. We construct an infinite sequence of positive integers by repeatedly appending, in order, one at a time, the digits from the list D to the integer k, in one of four ways: always on the left, always on the right, alternating and starting on the left, or alternating and starting on the right.(More)
In 1990 the second author constructed a basis for the centre of the Hecke algebra of the symmetric group S n over Q[ξ] using norms [13]. An integral " minimal " basis was later given by the first author in 1999 [5], following [9]. In principle one can then write elements of the norm basis as integral linear combinations of minimal basis elements. In this(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 Jucys–Murphy elements. We also discuss the existence of integral bases for the centre of the Hecke algebra that consist solely of monomial symmetric polynomials of Jucys–Murphy(More)
Using an idea of Erd˝ os, Sierpi´nski proved that there exist infinitely many odd positive integers k such that k · 2 n + 1 is composite for all positive integers n. In this paper we give a brief discussion of Sierpi´nski's theorem and some variations that have been examined, including the work of Riesel, Brier, Chen, and most recently, Filaseta, Finch and(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 ≡ 3 (mod 4), occur with even exponent in the prime factorization of z. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the(More)