Michael Filaseta

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In this paper, the authors continue their work on the problem of nding an h = h(x) as small as possible such that for x su ciently large, there is a squarefree number in the interval (x; x + h]: This problem has been investigated by Fogels [4], Roth [11], Richert [10], Rankin [9], Schmidt [12], Graham and Kolesnik [5], the second author [14,15], and the rst(More)
We address conjectures of P. Erdős and conjectures of Y.-G. Chen concerning the numbers in the title. We obtain a variety of related results, including a new smallest positive integer that is simultaneously a Sierpiński number and a Riesel number and a proof that for every positive integer r, there is an integer k such that the numbers k, k2, k3, . . . , kr(More)
Let and N be positive real numbers, and let f : R 7! R be any function. In this paper, we will obtain estimates for the size of the set fu 2 (N; 2N ] : jjf(u)jj < g, where u represents an integer and jjf(u)jj represents the distance from f(u) to the nearest integer. Thus, the set consists of u 2 (N; 2N ] for which ff(u)g, the fractional part of f(u), lies(More)