{"id":"hexagon/2610.00407","pdf":"https://hexagonmath.org/pdf/2610.00407v1","url":"https://hexagonmath.org/abs/2610.00407v1","tags":["msc:11D68","egyptian fractions","unit fractions"],"score":0,"title":"Optimal bounds for the length and number of unit fraction representations","domain":"math.NT","source":"hexagon","authors":["Thomas Bloom","Wouter van Doorn"],"reviews":0,"subject":"math.NT","summary":"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 $k$.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}