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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)
In 1849, Alphonse de Polignac conjectured that every odd positive integer can be written in the form 2n + p, for some integer n ≥ 0 and some prime p. In 1950, Erdős constructed infinitely many counterexamples to Polignac’s conjecture. In this article, we show that there exist infinitely many positive integers that cannot be written in either of the forms Fn(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)
In 1960, Sierpiński proved that there exist infinitely many odd positive rational integers k such that k · 2n + 1 is composite in Z for all n ≥ 1. Any such integer k is now known as a Sierpiński number, and the smallest value of k produced by Sierpiński’s proof is k = 15511380746462593381. In 1962, John Selfridge showed that k = 78557 is also a Sierpiński(More)
A unit x in a commutative ring R with identity is called exceptional if 1−x is also a unit in R. For any integer n ≥ 2, define φe(n) to be the number of exceptional units in the ring of integers modulo n. Following work of Shapiro, Mills, Catlin and Noe on iterations of Euler’s φ-function, we develop analogous results on iterations of the function φe, when(More)