On the d-representation of integers

Abstract

Let [Formula: see text] be two coprime positive integers. A positive integer [Formula: see text] is said to be [Formula: see text]-representable for [Formula: see text] if [Formula: see text] is the sum of terms taken from [Formula: see text] such that no one divides the other. Let [Formula: see text] denote the set of all positive integers that are [Formula: see text]-representable. For [Formula: see text], let [Formula: see text] be the maximum of the least terms of [Formula: see text]-representations of [Formula: see text] and let [Formula: see text] be the minimum of the largest terms of [Formula: see text]-representations of [Formula: see text]. In 1996, Erdős and Lewin conjectured that [Formula: see text] as [Formula: see text]. Recently, the authors of this paper confirmed this conjecture. This paper concerns the magnitudes of [Formula: see text] and [Formula: see text].