A question of Erdős and Graham on Egyptian fractions

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 $x$.

💡 Research Summary

The paper resolves a long‑standing question of Erdős and Graham concerning the number of Egyptian fraction representations with bounded denominators. For any fixed positive rational number (x), the authors prove that the number of subsets (A\subseteq