Short Egyptian fractions and doubly exponential families

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Family 025 · Number theory
Importance 3/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

An Egyptian fraction writes a positive rational number as a sum of distinct reciprocals of positive integers. For example,

$$\frac23=\frac12+\frac16,\qquad 1=\frac12+\frac13+\frac16.$$

Every positive rational number has such an expansion, but how many terms are needed in the worst case? If the denominator of the original fraction is $b$, can every numerator $1\le a<b$ be handled using only a constant multiple of $\log\log b$ terms?

Family 025 claims this conjecture of Erdős. It also derives two striking consequences: the number of distinct-term expansions of one grows doubly exponentially with the allowed exact length, and every integer below a doubly exponential threshold can occur as an exact denominator in some expansion of that length.

The length is optimized without any prescribed upper bound on the new denominators. That tradeoff is essential to understanding the result.

2. Background and essential definitions

A unit fraction has numerator one. For integers $1\le a<b$, define $N(a,b)$ to be the least $k$ for which

$$\frac ab=\frac1{n_1}+\cdots+\frac1{n_k}, \qquad 2\le n_1<\cdots<n_k.$$

The denominators are positive integers and must be distinct. They are written in increasing order so that permuting terms does not produce a new representation. The original fraction need not be reduced.

The worst numerator at denominator $b$ is measured by

$$N(b)=\max_{1\le a<b}N(a,b).$$

This is a maximum of minimum lengths. It does not say every fraction needs that many terms: $N(1,b)=1$ for all $b\ge2$.

The greedy algorithm repeatedly subtracts the largest unit fraction no greater than the current remainder. If the remainder is $a/b$, it chooses denominator $\lceil b/a\rceil$. This proves existence because the numerator of the new remainder decreases after reduction, but existence alone does not supply the desired uniform length bound.

For expansions of one, let

$$F(k)=\#\left\{(n_1,\ldots,n_k): 1\le n_1<\cdots<n_k,\ \sum_{i=1}^{k}\frac1{n_i}=1\right\}.$$

The length is exactly $k$. For $k>1$, no denominator can be one, since its reciprocal already accounts for the whole sum.

Let $D_k$ be the set of integers $m\ge2$ occurring as a denominator in at least one such expansion, and define

$$v(k)=\min\bigl(\{2,3,\ldots\}\setminus D_k\bigr).$$

Each $m\in D_k$ may use a different expansion. The statement that every $m<v(k)$ occurs does not mean one expansion contains all those denominators.

“Doubly exponential in $k$” means growth comparable on the double-logarithmic scale: $\log\log F(k)$ or $\log\log v(k)$ is bounded above and below by positive multiples of $k$. It is stronger than ordinary exponential growth and does not specify one exact exponential base.

3. Development history and previous partial results

The representation itself is ancient, but the modern problem concerns uniform bounds. Nakayama studied arithmetic criteria for short expansions. In 1950, Erdős improved a bound reported from de Bruijn to order $\log b/\log\log b$, proposed the $O(\log\log b)$ upper bound, and proved a matching lower order already using the numerator $b-1$.

Vose’s 1985 paper improved the uniform upper bound to order $\sqrt{\log b}$. The double-logarithmic scale remained the conjectural target. The new manuscript’s lower bound is classical; its claimed advance is the matching upper bound for every numerator and every sufficiently large denominator.

A related problem restricts denominator sizes as well as length. Tenenbaum and Yokota obtained a length at most $(1+\varepsilon)\log b/\log\log b$, for fixed $\varepsilon>0$ and large $b$, while keeping every denominator at most $4b(\log b)^2\log\log b$. The present theorem optimizes only length and should not be read as improving both constraints simultaneously.

Erdős and Graham also asked how many expansions of one have a given length, and which denominators can appear. Konyagin obtained a lower bound for the count whose double logarithm has order $k/\log k$. Elsholtz obtained this order even with specified congruence restrictions on denominators. Elsholtz–Planitzer established an upper bound with double logarithm $O(k)$, including a setting that allows repetitions.

For prescribed denominators, van Doorn and Tang proved $v(k)\ge\exp(ck^2)$ and developed operations that increase the length while preserving an exact chosen denominator. Their paper also anticipates the connection between uniform short expansions and a double-exponential threshold. Family 025 supplies the stronger length theorem and the required denominator-avoidance arguments.

4. What the new manuscripts claim

Theorem 1.1 states that absolute constants $c_1,c_2>0$ exist such that, for all sufficiently large integers $b$,

$$\boxed{c_1\log\log b\le N(b)\le c_2\log\log b.}$$

The upper bound applies to every integer numerator $1\le a<b$. The constants do not depend on that numerator or denominator. No upper bound on the resulting $n_i$ is prescribed.

Corollary 1.2 claims absolute constants $c,C>0$ with

$$ck\le\log\log F(k)\le Ck$$

for every sufficiently large integer $k$. In particular, the conclusion concerns all sufficiently large exact lengths, not merely an infinite subsequence of lengths.

Corollary 1.3 claims

$$\exp(\exp(k/600))\le v(k)\le1+k^{2^{k-1}}$$

for sufficiently large $k$, and gives the sharper slope bounds

$$\frac{\log2}{257} \le\liminf_{k\to\infty}\frac{\log\log v(k)}k \le\limsup_{k\to\infty}\frac{\log\log v(k)}k \le\log2.$$

Equivalently, every fixed $c<\log2/257$ eventually gives $v(k)\ge\exp(\exp(ck))$. These inequalities determine the double-logarithmic order, not an exact asymptotic coefficient. The convenient elementary upper bound is not presented as the sharpest available public upper estimate.

The prescribed-denominator conclusion retains the exact term $1/m$. Merely keeping a denominator divisible by $m$ would not satisfy it. The counting consequence needs a weaker divisibility feature, so the two corollaries require different constructions.

5. Why the advance matters

Provisional importance: 3/5. Taken as stated, the result reaches the optimal order in a longstanding uniform Egyptian-fraction problem. Its elementary formulation conceals a difficult demand: no numerator may be left in an exceptional set.

The consequences quantify how much flexibility remains among exact-length expansions of one. Short representations enable both an enormous number of choices and inclusion of every moderately sized prescribed denominator, on a doubly exponential scale.

The advance is substantial within additive and combinatorial number theory. It does not settle all famous fixed-length problems, such as asking for a particular three-term pattern for every denominator, and it makes no simultaneous claim that the new denominators are small.

6. The main idea

First solve almost every numerator over a useful denominator

Set $S=\log b$. A short greedy prefix either finishes the expansion or leaves a remainder $A/C$, with $A\le b$ and $C$ in a controlled exponential range in $S$.

The manuscript then constructs an auxiliary integer $M$, also exponential in $S$, for which a dense set of numerators $u$ gives short expansions of $u/(MC)$. “Dense” here is strong enough that a suitable multiple $gAM$ can be written as the sum of two good numerators. Adding their expansions and dividing every term by $g$ produces an expansion of the original remainder $A/C$.

This is the passage from almost every numerator in a carefully designed interval to every original numerator. A probabilistic or density result alone would not establish the theorem without this final step.

Use divisors to reduce the numerator

For a divisor $t\mid M$, choose

$$z=\left\lceil\frac{Qt}{u}\right\rceil, \qquad h=uz-Qt,$$

where $Q$ is the current denominator factor. Then

$$\frac{u}{MQ} =\frac1{(M/t)z}+\frac1z\frac{h}{MQ}.$$

One unit fraction has been removed, and the residual numerator is $h$, a small nonnegative residue when $t$ is chosen well. The construction selects $M$ so that many of its divisors produce useful residues for most numerators at each scale.

Deterministic products of distinct primes handle larger numerator ranges, while independent random products supply estimates at smaller ranges. The paper descends through $O(\log S)$ ranges, changing the denominator factor from $C$ to one when appropriate. At the bottom, an all-numerator reduction and a binary expansion finish the process.

Keep exceptional sets from accumulating

Each stage may have bad numerators. It is not enough that each bad set is small separately: many earlier numerators could map into the same bad set later.

For fixed $t,h$, a predecessor $u$ must divide $Qt+h$. The manuscript proves a uniform moment bound for the number of small divisors in shifted intervals. Hölder’s inequality then controls how many predecessors a small exceptional set can have. This keeps the exceptional proportion small across the whole descent.

Intermediate lists can repeat denominators. A separate replacement argument removes repetitions while preserving the number of terms when the sum is below one. The proof uses identities replacing two equal reciprocals by two different ones and a finiteness argument to ensure termination. The expansion-of-one case receives its own treatment.

Produce many expansions, or preserve one exact denominator

For the counting corollary, short expansions are arranged to contain a denominator with many prime factors. Splitting its reciprocal using its divisors creates many distinct choices. An injective length-increasing operation extends the count to every sufficiently large prescribed length.

For the marker problem, reserve $1/m$ first and construct the rest while avoiding that denominator. A greedy prefix is chosen so the remaining positive sum is smaller than every reserved reciprocal. Any positive unit-fraction expansion of that remainder must then use larger denominators, so it cannot collide with the reserved terms.

A separate padding lemma increases the length without losing $1/m$. A direct quantitative construction gives the coefficient $257/\log2$ in the sufficient length bound, hence the explicit lower slope for $\log\log v(k)$. It does not require making the unspecified constant in the general length theorem explicit.

7. Scope, limitations, and open questions

The optimal-order statement concerns $N(b)$, the worst numerator. Many individual fractions have much shorter expansions. It does not claim that each numerator needs $\Theta(\log\log b)$ terms.

The denominators in the expansion can be large, and the theorem gives no efficient algorithm or practical runtime bound for finding a shortest expansion. Controlling length and denominator size simultaneously remains a separate optimization problem.

The count $F(k)$ uses distinct positive denominators and exact length. The bounds do not identify a limiting value of $\log\log F(k)/k$. Likewise, the lower and upper slopes for $v(k)$ are not equal, so its precise doubly exponential growth is still undetermined.

Finally, denominator coverage is a collection of existence statements, one for each $m$. It does not demand that a single $k$-term representation contain all integers below the threshold.

8. Sources and further reading

Theorem 1.1 gives the uniform length bound, and Corollaries 1.2–1.3 give the counting and prescribed-denominator consequences. Sections 2–5 explain the dense-family reduction, divisor moments and multiscale descent. Sections 6–8 distinguish counting, exact-marker preservation and the quantitative marker construction.

Vose and van Doorn–Tang are linked in section 3. The manuscript’s bibliography identifies Erdős’s original theorem and conjecture, Tenenbaum–Yokota’s simultaneous denominator bounds, and the earlier counting results. Optional background on Egyptian fractions gives further examples and classical algorithms.

Source manuscripts

• M1. OpenAI, Short Egyptian fractions (September 25, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.