A lower bound for the number of Egyptian fractions

An Egyptian fraction is a sum of the form 1 / n 1 + ⋯ + 1 / n r 1/n_{1} + \cdots + 1/n_{r} where n 1 n_{1} , …, n r n_{r} are distinct positive integers. We prove explicit lower bounds for the cardinality of the set E N {E}_{N} of rational numbers that can be represented by Egyptian fractions with denominators not exceeding N N . More precisely, we show that for every integer k ≥ 4 k \geq 4 such that ln k ⁡ N ≥ 3 / 2 \ln _{k} N \geq 3/2 it holds ln ⁡ | E N | ln ⁡ 2 ≥ ( 2