HEXAGONbeta

Optimal bounds for the length and number of unit fraction representations

Primarily human-written textHuman understanding: all partsmath.NT — Number Theory

Contributed by Thomas Bloom ↗, Wouter van Doorn

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

This paper gives a human-written exposition and simplification of proofs that were recently posted by OpenAI. We prove that every a/b∈(0,1)a/b\in (0, 1) can be written as the sum of O(log⁡log⁡b)O(\log\log b) many distinct unit fractions. We also prove that the number of representations of 11 as the sum of kk distinct unit fractions grows doubly exponentially in kk.

Provenance statement

This is an entirely human written exposition (and partial simplification) of AI-generated proofs posted by OpenAI on 06/10/2026.

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Version history

  1. v1Submitted by Thomas BloomInitial depositCurrentOct 20, 2026