Counting Egyptian fractions

Carlo Sanna; Giuseppe Molteni; Lo\"ic Greni\'e; Sandro Bettin

arxiv: 1906.11986 · v2 · pith:N5E3DZKQnew · submitted 2019-06-27 · 🧮 math.NT

Counting Egyptian fractions

Sandro Bettin , Lo\"ic Greni\'e , Giuseppe Molteni , Carlo Sanna This is my paper

Pith reviewed 2026-05-25 14:19 UTC · model grok-4.3

classification 🧮 math.NT

keywords egyptian fractionsunit fractionscardinality boundsiterated logarithmssums of reciprocalsnumber theorydenominator restriction

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The pith

Egyptian fractions with denominators at most N have log-cardinality between N/log N times iterated logs and 0.421 N.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the set of all Egyptian fractions formed from unit fractions whose denominators are at most N and derives explicit bounds on the logarithm of the size of this set. It proves a lower bound that grows like N over log N multiplied by any fixed number of iterated logarithms, together with an upper bound linear in N. These inequalities hold for every sufficiently large N once the number of iterated logarithms is fixed. A reader cares because the result quantifies how many distinct sums of distinct reciprocals are possible under a denominator restriction, which controls the richness of Egyptian-fraction representations.

Core claim

For the set E_N of Egyptian fractions using denominators at most N, the cardinality satisfies N / log N times the product of iterated logarithms from 3 to k is less than log of the cardinality, which is itself less than 0.421 N, for any fixed k at least 3 and all sufficiently large N.

What carries the argument

The set E_N of all finite sums of distinct unit fractions with denominators ≤ N, together with the two-sided bounds on the logarithm of its cardinality.

Load-bearing premise

The stated inequalities hold for every sufficiently large N once the number of iterated logarithms is fixed at any integer k at least 3.

What would settle it

An explicit N large enough that either log of the number of such fractions falls below the lower bound or exceeds 0.421 N would disprove the claim.

Figures

Figures reproduced from arXiv: 1906.11986 by Carlo Sanna, Giuseppe Molteni, Lo\"ic Greni\'e, Sandro Bettin.

**Figure 1.** Figure 1: Graph of log(#EN )/N (dots) and of log(#EN )/(N/ log N) (triangles) for N ≤ 43. Acknowledgements. S. Bettin is member of the INdAM group GNAMPA. L. Greni´e, G. Molteni and C. Sanna are members of the INdAM group GNSAGA. C. Sanna is supported by a postdoctoral fellowship of INdAM. The extensive computations needed for this paper have been performed on the UNITECH INDACO computing platform of the Universit… view at source ↗

read the original abstract

For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$ \frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N, $$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit bounds on log of the count of Egyptian fractions with denominators <=N: lower bound with N/log N times iterated logs, upper bound 0.421 N.

read the letter

The main point is that this paper proves explicit bounds on the logarithm of the number of distinct Egyptian fractions using denominators at most N. The upper bound is 0.421 N and the lower bound takes the form N over log N times a product of iterated logarithms from 3 up to any fixed k at least 3, valid for all large enough N. These are presented as proven estimates rather than existence results. What stands out as new is the concrete numerical coefficient on the upper side and the explicit iterated-log structure on the lower side, which go beyond the vaguer statements one often sees for subset-sum cardinalities in this setting. The work does a clean job of stating the result in a form that can be used directly. The bounds are consistent with the fact that there are only 2^N possible subsets but many sums collide, so an upper coefficient well below log 2 is expected. Soft spots are minor. The constant 0.421 is not asserted to be best possible, and the lower bound requires choosing k first, which is standard but means the explicit lower estimate weakens as more logs are included. The proofs themselves are not visible in the abstract, so one cannot check the analytic estimates or error terms in detail, but the statement shows no circularity or internal inconsistency. This is a specialized result in analytic number theory aimed at people working on Egyptian fractions or harmonic subset sums. A reader who needs quantitative control on the size of this set will find it useful. It deserves a serious referee because the claims are concrete, the scope is clearly stated, and the problem is well-posed even if the broader impact stays narrow.

Referee Report

0 major / 3 minor

Summary. The manuscript defines 𝔈_N as the set of all sums of distinct unit fractions 1/n with n ≤ N. It proves that for any fixed integer k ≥ 3 and all sufficiently large N, (N / log N) ∏_{j=3}^k log_j N < log(#𝔈_N) < 0.421 N, where log_j denotes the j-th iterated logarithm.

Significance. The bounds quantify the number of distinct subset sums of the first N harmonic terms. The upper bound improves on the trivial 2^N by establishing a linear exponent with coefficient 0.421 < log 2, while the lower bound shows that the cardinality is at least exp(Ω(N / log N ⋅ iterated-log product)) for arbitrary fixed depth. This contributes to additive combinatorics by controlling collisions among harmonic sums.

minor comments (3)

[Abstract] Abstract: the iterated-log product begins at j=3; a brief parenthetical definition of log_j would improve immediate readability.
[Theorem 1.2 (or equivalent)] The upper-bound constant 0.421 is stated without an explicit reference to the optimization or inequality used to obtain it; adding a short derivation sketch or citation in the statement of the main theorem would help.
[Introduction] Notation: ensure that the symbol 𝔈_N is introduced with the same font and fraktur style in both the abstract and the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript, their assessment of its significance in additive combinatorics, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; bounds derived from subset-sum structure

full rationale

The paper states and proves asymptotic bounds on log(#E_N) for the cardinality of distinct subset sums of {1/n : n ≤ N}. The lower bound grows like N/log N times a fixed-depth product of iterated logarithms (valid for any fixed k ≥ 3 and large N), while the upper bound is linear in N with coefficient < log 2. These are standard analytic-combinatorial estimates on the number of distinct harmonic subset sums; they do not reduce to fitted parameters renamed as predictions, self-definitions, or load-bearing self-citations. The derivation chain relies on external number-theoretic tools (e.g., estimates on harmonic sums and subset-sum collisions) rather than tautological restatements of the input set E_N itself. No step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard definition of Egyptian fractions and analytic number theory tools for counting distinct sums; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

domain assumption Egyptian fractions are finite sums of distinct positive unit fractions with denominators ≤ N
This defines the set E_N whose cardinality is bounded.

pith-pipeline@v0.9.0 · 5630 in / 1104 out tokens · 43758 ms · 2026-05-25T14:19:24.339535+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We give upper and lower bounds for the cardinality of E_N, proving that N / log N * product from j=3 to k of log_j N < log(#E_N) < 0.421 N
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2. We have α<0.421, that is, log(#EN)<0.421N for all N large enough.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Bettin, G

S. Bettin, G. Molteni, and C. Sanna, Greedy approximations by signed harmonic sums and the Thue– Morse sequence, https://arxiv.org/abs/1805.00075

work page arXiv
[2]

Bettin, G

S. Bettin, G. Molteni, and C. Sanna, Small values of signed harmonic sums , C. R. Math. Acad. Sci. Paris 356 (2018), no. 11-12, 1062–1074

work page 2018
[3]

M. N. Bleicher and P. Erd˝ os, The number of distinct subsums of ∑N 1 1/i, Math. Comp. 29 (1975), 29–42, Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday

work page 1975
[4]

E. S. Croot, III, On some questions of Erd˝ os and Graham about Egyptian fractions, Mathematika 46 (1999), no. 2, 359–372

work page 1999
[5]

Martin, Dense Egyptian fractions , Trans

G. Martin, Dense Egyptian fractions , Trans. Amer. Math. Soc. 351 (1999), no. 9, 3641–3657

work page 1999
[6]

Martin, Denser Egyptian fractions , Acta Arith

G. Martin, Denser Egyptian fractions , Acta Arith. 95 (2000), no. 3, 231–260

work page 2000
[7]

Rosser, Explicit bounds for some functions of prime numbers , Amer

B. Rosser, Explicit bounds for some functions of prime numbers , Amer. J. Math. 63 (1941), 211–232

work page 1941
[8]

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences , http://oeis.org

work page
[9]

Tenenbaum and H

G. Tenenbaum and H. Yokota, Length and denominators of Egyptian fractions. III , J. Number Theory 35 (1990), no. 2, 150–156

work page 1990
[10]

M. D. Vose, Egyptian fractions , Bull. London Math. Soc. 17 (1985), no. 1, 21–24

work page 1985
[11]

Yokota, On a conjecture of M

H. Yokota, On a conjecture of M. N. Bleicher and P. Erd˝ os , J. Number Theory 24 (1986), no. 1, 89–94

work page 1986
[12]

Yokota, On a problem of Bleicher and Erd˝ os , J

H. Yokota, On a problem of Bleicher and Erd˝ os , J. Number Theory 30 (1988), no. 2, 198–207. COUNTING EGYPTIAN FRACTIONS 11 Dipartimento di Matematica, Universit `a di Genova, Via Dodecaneso 35, 16146 Genova, Italy E-mail address : bettin@dima.unige.it Dipartimento di Ingegneria Gestionale, dell’Informazion e e della Produzione, Universit `a di Bergamo, ...

work page 1988

[1] [1]

Bettin, G

S. Bettin, G. Molteni, and C. Sanna, Greedy approximations by signed harmonic sums and the Thue– Morse sequence, https://arxiv.org/abs/1805.00075

work page arXiv

[2] [2]

Bettin, G

S. Bettin, G. Molteni, and C. Sanna, Small values of signed harmonic sums , C. R. Math. Acad. Sci. Paris 356 (2018), no. 11-12, 1062–1074

work page 2018

[3] [3]

M. N. Bleicher and P. Erd˝ os, The number of distinct subsums of ∑N 1 1/i, Math. Comp. 29 (1975), 29–42, Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday

work page 1975

[4] [4]

E. S. Croot, III, On some questions of Erd˝ os and Graham about Egyptian fractions, Mathematika 46 (1999), no. 2, 359–372

work page 1999

[5] [5]

Martin, Dense Egyptian fractions , Trans

G. Martin, Dense Egyptian fractions , Trans. Amer. Math. Soc. 351 (1999), no. 9, 3641–3657

work page 1999

[6] [6]

Martin, Denser Egyptian fractions , Acta Arith

G. Martin, Denser Egyptian fractions , Acta Arith. 95 (2000), no. 3, 231–260

work page 2000

[7] [7]

Rosser, Explicit bounds for some functions of prime numbers , Amer

B. Rosser, Explicit bounds for some functions of prime numbers , Amer. J. Math. 63 (1941), 211–232

work page 1941

[8] [8]

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences , http://oeis.org

work page

[9] [9]

Tenenbaum and H

G. Tenenbaum and H. Yokota, Length and denominators of Egyptian fractions. III , J. Number Theory 35 (1990), no. 2, 150–156

work page 1990

[10] [10]

M. D. Vose, Egyptian fractions , Bull. London Math. Soc. 17 (1985), no. 1, 21–24

work page 1985

[11] [11]

Yokota, On a conjecture of M

H. Yokota, On a conjecture of M. N. Bleicher and P. Erd˝ os , J. Number Theory 24 (1986), no. 1, 89–94

work page 1986

[12] [12]

Yokota, On a problem of Bleicher and Erd˝ os , J

H. Yokota, On a problem of Bleicher and Erd˝ os , J. Number Theory 30 (1988), no. 2, 198–207. COUNTING EGYPTIAN FRACTIONS 11 Dipartimento di Matematica, Universit `a di Genova, Via Dodecaneso 35, 16146 Genova, Italy E-mail address : bettin@dima.unige.it Dipartimento di Ingegneria Gestionale, dell’Informazion e e della Produzione, Universit `a di Bergamo, ...

work page 1988