Short Egyptian fractions

1 · The essentials

A fraction strictly between zero and one can be written as a sum of fractions whose numerators are all one. The main result says this can be done with few terms and without repeating any denominator. For a sufficiently large starting denominator $b$ , the number of terms needed is at most a fixed constant times $lo g lo g b$ . This quantity grows very slowly with $b$ . If the guarantee must work for every numerator, this order of growth cannot be improved. The manuscript thus resolves a conjecture of Erdős about these short representations, and the main result has a Lean formalization. It also studies how many such representations add up to exactly one, and which denominators can occur when the number of terms is fixed.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine portions measuring one half, one third, one quarter, and so on, with only one portion of each size available. The result says that any rational amount strictly between zero and one can be assembled using few of these portions; the image alone does not explain the precise bound.

Why it matters

The result does more than guarantee a representation exists: it controls its length uniformly over all allowed numerators. The resulting bound has the right order of growth in the worst case. The results about representations of one complement this study of length by examining their number and the denominators missing from them.

2 · Going further

A positive unit fraction is a number of the form $1/ n$ , where $n$ is a positive integer. The problem is to write $a / b = \sum_{i = 1}^{k} 1/ n_{i}$ , with all the $n_{i}$ distinct, while keeping $k$ as small as possible. For every sufficiently large integer $b$ and every integer $a$ with $1 \leq a < b$ , the main result gives $k \leq C lo g lo g b$ , where $C$ is an absolute constant.

The manuscript presents this result as resolving a conjecture of Erdős. The order $lo g lo g b$ is optimal when the numerator varies: this is a worst-case statement over numerators, not a claim that every fraction requires that many terms. The supplied abstract describes neither the construction of the denominators nor the steps of the argument, so a method cannot be reconstructed from this source alone.

The same manuscript also shows two results about representations of $1$ with exactly $k$ distinct positive unit fractions. Both their number and the least integer at least $2$ that never appears as a denominator in such a representation grow doubly exponentially in $k$ : their double logarithms have order $k$ . No companion manuscript is supplied here; these two results appear in the main manuscript’s abstract.

Useful notions

Fractions and rational numbers
Sums of fractions
Logarithms and repeated logarithms
Orders of growth

What was known before / what is new

Before

The source presents the short-representation question as a conjecture of Erdős. It gives neither earlier bounds nor previously known methods, and supplies no history for the two questions about representations of one.

New

The main result establishes a uniform bound of $O (lo g lo g b)$ distinct terms with an absolute constant, and this order is optimal when the numerator varies. The manuscript also shows doubly exponential growth for the number of representations of one with exactly $k$ terms and for the first denominator absent from all such representations.

Limits and assumptions

The precise statement in the abstract concerns sufficiently large integers $b$ and $1 \leq a < b$ ; the unit fractions must be positive and distinct. Optimality concerns the worst case as the numerator varies, not each individual fraction. The main short-representation result has a Lean formalization; the supplied information does not specify whether it also covers the two additional results. The source gives no explicit constant, explicit threshold, or description of the argument, and identifies no remaining open question.

Mini glossary

Positive unit fraction: A fraction of the form $1/ n$ , where $n$ is a positive integer.
Egyptian fraction representation: A representation of a number as a sum of distinct positive unit fractions.
Absolute constant: A constant that depends on neither the numerator $a$ nor the denominator $b$ .
$O (lo g lo g b)$ bound: An upper bound equal to a fixed constant times $lo g lo g b$ for sufficiently large $b$ .
Doubly exponential growth: In the results described here, growth of a quantity whose double logarithm lies between fixed positive multiples of $k$ for sufficiently large $k$ .

Keywords : Egyptian fractions · Erdős’s conjecture · Double logarithm

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/025.md

Manuscripts in this family

★

Short Egyptian fractions

25/09/2026

Verified by Lean

We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number $a / b$ with $1 \leq a < b$ is a sum of $O (lo g lo g b)$ 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 GitHub folder Lean formalization