A question of Erdős and Graham on Egyptian fractions

Abstract

Answering a question of Erdős and Graham, we show that for each fixed positive rational number $x$ the number of ways to write $x$ as a sum of reciprocals of distinct positive integers each at most $n$ is $2^{(c_x+o(1))n}$ for an explicit constant $c_x$ increasing with $x$.

Reformatted for Discrete Analysis