Binary Egyptian Fractions

Classroom note: Ahmes expansions of rational numbers of length two

“…In particular, Propositions 3.1-3.3 establish that 'most' rational numbers of the form 2/n, 3/n, or 4/n can be expressed as a sum of two distinct unit fractions. This stands in sharp contrast to the upshot of [4] noted earlier that the probability (based on natural density) is zero that a 'random' proper positive rational number is expressible as a sum of two distinct unit fractions. As usual, let N denote the set of positive integers.…”

“…A standard estimate for the totient function [14,Theorem A.16] ensures that there exists an integer N ¼ Nð"Þ > 0 such that, for all n > N, we have n 1À"=2 < 'ðnÞ < n. Moreover, we know, as a consequence of [4,Corollary 3], that lim n!1 B 2 ðnÞ=n ¼ 0 for all > 0. Now, if n > Nð"Þ, then 0 < B 2 ðnÞ 'ðnÞ " < B 2 ðnÞ n ð1Àð"=2ÞÞ"…”

“…We know, as a consequence of [4,Proposition 10], that there exists an essentially computable rational number k such that (i) 0 4 k < 1 and (ii) for each " > 0, there exists D > 0 such that B k ðnÞ 4 Dn k þ" . Fix " such that 0 < " < 1 À k .…”

“…Fix any positive integer k. By analogy with the notation in [4], for each positive integer n, we define integers B k ðnÞ :¼ a 2 N j 1 4 a 4 n; ða; nÞ ¼ 1;…”

“…In particular, Sierpin´ski [2, The´ore`me 2] demonstrated that the set of all positive rational numbers with an expansion involving at most k unit fractions is nowhere dense in the set of all positive rational numbers. More recently, Croot et al [4,Proposition 10] showed that for each " > 0 and k > 1, the number of positive rational numbers a/n admitting an expansion involving precisely k unit fractions is ( k;" n k þ" as a function of n, where k is a rational number determined by a recurrence relation and 0 4 k < 1. The upshot of this result is that the natural density (in the sense of [1, p. 241]) of the proper positive rational numbers just mentioned is 0.…”