Result 025, Number theory

Short Egyptian fractions

Every rational $a/b$ with $1\le a\lt b$ is a sum of $O(\log\log b)$ distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.

Lean formalization Proof

The bigger picture

Why it matters

How many fractions with numerator 1 does it take to express a rational number? The manuscript claims a sharp answer: even as the denominator grows, the worst-case number of terms grows extremely slowly.

What changes?

An Egyptian fraction expansion is a sum of reciprocals of distinct positive integers. The manuscript reports that, for every sufficiently large integer b and every integer a between 1 and b minus 1, a/b has such an expansion with at most an absolute constant times log log b terms. That constant is independent of a and b. As the numerator varies, the worst-case minimum number of terms has the same order, settling the stated Erdős conjecture.

What does that help mathematicians do?

The matching lower bound rules out a uniformly smaller growth rate for the number of terms, while allowing particular fractions to have shorter expansions. The manuscript also quantifies expansions of 1 with exactly k distinct terms: both their number and the smallest integer at least 2 excluded from all their denominators grow doubly exponentially, with double logarithms of order k. These claims measure both the abundance of expansions and how comprehensively their denominators cover small integers.

Are there practical applications?

The immediate value is foundational: the result identifies the optimal scale for representing rational numbers using distinct unit fractions. It supplies a sharp benchmark for work on constructing such representations. The abstract gives existence and counting claims, however, not runtime guarantees for finding an expansion or bounds on the sizes of its denominators.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Short Egyptian fractions

September 25, 2026 33 pages Main result formalized in Lean

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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Short-Egyptian-fractions-September-25-2026,
  author = {{OpenAI}},
  title = {{Short Egyptian fractions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf}{OAI:Short-Egyptian-fractions-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/025.md.

Short Egyptian fractions

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Short Egyptian fractions

Scope

An Egyptian-fraction expansion writes a rational number as a sum of distinct unit fractions. The formalization proves that every $a/b$ with $1\le a<b$ has such an expansion and that the largest minimum length at denominator $b$ is $\Theta(\log\log b)$ . If $F(k)$ counts exact $k$ -term expansions of $1$ , it also proves $\log\log F(k)=\Theta(k)$ and bounds every denominator in such an expansion.

Every integer $m\ge2$ occurs as a denominator in an expansion of $1$ , with length eventually at most $(257/\log2+\varepsilon)\log\log m$ for every $\varepsilon>0$ ; padding can preserve that denominator. If $v(k)$ is the least integer at least $2$ absent from all exact $k$ -term expansions of $1$ , then eventually $e^{e^{k/600}}\le v(k)\le1+k^{2^{k-1}}$ , and $\liminf_{k\to\infty}\log\log v(k)/k\ge\log2/257$ .

Comparator links

Result	Comparator statement
Short expansions, counting, and prescribed denominators	EgyptianFractions.lean
Optimal order of the shortest Egyptian-fraction expansions	ShortEgyptianFractions.lean

At a glance

Subject: Number theory
Manuscripts: 1
Pages: 33
Dated: Sep 25, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Erdős’s conjecture on short Egyptian fractionsProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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By similarity of summaries and abstracts.

On the map

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