math.NT — Number Theory

Optimal bounds for the length and number of unit fraction representations

Contributed by Thomas Bloom, Wouter van Doorn

This paper gives a human-written exposition and simplification of proofs that were recently posted by OpenAI. We prove that every $a/b\in (0, 1)$ can be written as the sum of $O(\log\log b)$ many distinct unit fractions. We also prove that the number of representations of $1$ as the sum of $k$ distinct unit fractions grows doubly exponentially in $k$.