Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 025-01. Short Egyptian fractions
Manuscript map · lines 2945–2954
OpenAI@2026-10-07#rem:oai-p-025-01
Authors: OpenAI. Family 025.
Title: Short Egyptian fractions.
Abstract: We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number $a/b$ with $1{}le a{}lt b$ is a sum of $O({}log{}log b)$ distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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