The strange story of an almost unknown prime number counter: The Rafael Barrett formula

Eduardo Mizraji

arxiv: 2509.19324 · v6 · submitted 2025-09-12 · 🧮 math.HO

The strange story of an almost unknown prime number counter: The Rafael Barrett formula

Eduardo Mizraji This is my paper

Pith reviewed 2026-05-18 17:36 UTC · model grok-4.3

classification 🧮 math.HO

keywords prime number counterRafael Barrett1903 formulaHenri Poincaré1935 publicationMontevideo journalprime distributionhistorical mathematics

0 comments

The pith

Rafael Barrett created a prime-counting formula in 1903 that stayed unknown until its 1935 publication.

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

The paper recounts the creation of a prime number counter by Rafael Barrett in 1903 as part of a note sent to Henri Poincaré. This formula remained undiscovered for decades until a Uruguayan mathematician found it in the 1930s and arranged for its publication and analysis in a Montevideo journal in 1935. The study presents the formula and examines a challenge that it may pose to existing views on prime distribution. A sympathetic reader would care because the episode shows how mathematical ideas can vanish from view and then reappear, supplying a possible new tool for counting primes.

Core claim

Barrett's formula, created in 1903, functions as a prime number counter and was published in 1935 after being discovered in the 1930s, posing a challenge worth analyzing.

What carries the argument

Barrett's prime number counter formula, an expression that tallies the primes up to a chosen limit.

If this is right

If valid, the formula supplies an alternative method for counting primes up to a given number.
The 1935 publication made the method available to readers in South America for the first time.
Further analysis of the formula could clarify aspects of prime distribution that standard approaches leave open.
The episode illustrates how private correspondence can preserve mathematical results that later surface in print.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Archival searches of early twentieth-century correspondence might uncover other overlooked prime-counting expressions.
Numerical tests on modern computers could check how closely the formula matches known prime counts for large limits.
Placing the formula beside other early attempts at prime counters might reveal shared ideas or independent lines of thought.

Load-bearing premise

The historical facts about the formula's creation in 1903, its long period of obscurity, and its 1935 publication are accurate and the formula is presented without transcription errors.

What would settle it

Archival evidence showing the formula was known or published before 1935, or a direct check proving the formula fails to count primes correctly for a tested range of values.

Figures

Figures reproduced from arXiv: 2509.19324 by Eduardo Mizraji.

**Figure 3.** Figure 3: shows what Rafael Barrett prepared for Poincaré taken from the article by García de Zúñiga (who in 1935 was not sure if this proof had actually been sent) [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

read the original abstract

In this brief article, we present the formula created by Rafael Barrett in 1903 in a note to Henri Poincar\'e, which remained unknown for decades. Discovered in the 1930s by a Uruguayan mathematician, this formula was published and analyzed in a journal published in Montevideo in 1935. In this study, we present Barrett's formula and analyze a challenge it could pose.

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

This is a minor historical note recovering Barrett's obscure 1903 prime-counting formula and its 1935 publication.

read the letter

This paper is a short historical note about Rafael Barrett's 1903 formula for counting primes. Barrett sent it to Henri Poincaré, but it stayed unknown until a Uruguayan mathematician found it in the 1930s and published it in Montevideo in 1935. The author here presents the formula and looks at a challenge it might pose. What stands out is the recovery of this obscure work. The paper does well in outlining the timeline and the context around an early 20th century attempt at a prime counter. It highlights a contribution that might otherwise remain overlooked, especially one with ties to South American mathematics. If the transcription of the formula is faithful and the historical details are backed by proper references, this adds a legitimate footnote to the record of prime-counting ideas. The softer part is the mathematical analysis. The mention of a challenge is intriguing, but the paper does not appear to provide extensive verification, such as testing the formula against known values of the prime counting function or comparing it to contemporaries like those from Chebyshev or others. Without that, the challenge remains more suggestive than demonstrated. The work stays firmly in the history lane rather than venturing into new derivations. This kind of paper suits readers with an interest in the history of mathematics, particularly those studying the development of analytic number theory or the contributions of lesser-known figures. It won't appeal much to someone focused on current research in primes or computational methods. I think it deserves a serious referee. The topic is narrow, but if the sources hold up, it can be a useful addition to the literature on the history of these functions. Recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript is a brief historical note recounting that Rafael Barrett created a prime-number counting formula in 1903 and sent it to Henri Poincaré; the formula remained unknown for decades until its rediscovery in the 1930s by a Uruguayan mathematician, after which it was published and analyzed in a Montevideo journal in 1935. The paper presents the formula and examines a challenge it could pose.

Significance. If the historical timeline, correspondence details, and formula transcription are accurate, the note usefully documents an overlooked early contribution to prime-counting ideas from outside the main European mathematical centers. This may encourage further archival work on Barrett’s papers and on the circulation of mathematical ideas in Latin America during the early twentieth century.

major comments (1)

[Abstract and main text] The abstract states that the paper will 'analyze a challenge it could pose,' yet the manuscript supplies no explicit statement of what that challenge consists of, no comparison with the prime-number theorem or other known counting functions, and no indication of whether the 1935 analysis identified an error, an asymptotic discrepancy, or a computational advantage. This omission is load-bearing for the paper’s claim to mathematical interest.

minor comments (2)

Supply full bibliographic details (journal name, volume, pages, and exact title) for the 1935 Montevideo publication and, if possible, a reference or archival location for the 1903 note to Poincaré.
Ensure the formula is displayed as a numbered equation with all variables defined, so that readers can verify the transcription independently.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on the abstract and main text. We address the point below.

read point-by-point responses

Referee: [Abstract and main text] The abstract states that the paper will 'analyze a challenge it could pose,' yet the manuscript supplies no explicit statement of what that challenge consists of, no comparison with the prime-number theorem or other known counting functions, and no indication of whether the 1935 analysis identified an error, an asymptotic discrepancy, or a computational advantage. This omission is load-bearing for the paper’s claim to mathematical interest.

Authors: We agree that the current wording of the abstract creates an expectation of explicit analysis that the manuscript does not fully meet. The paper is intended as a brief historical note whose primary aim is to document the 1903 formula, its transmission to Poincaré, and its rediscovery and publication in Montevideo in 1935. The phrase 'analyze a challenge it could pose' was meant to refer to the historiographical and contextual implications highlighted in the 1935 journal discussion rather than a technical comparison with the prime-number theorem. Nevertheless, the referee is correct that this is not stated clearly. In the revised version we will replace the final sentence of the abstract with a more precise formulation and add one short paragraph after the formula is presented. This paragraph will quote or paraphrase the specific point raised in the 1935 analysis (as reported in the source) and note whether it concerned an error, an asymptotic difference, or a computational feature. No new mathematical derivations will be introduced; the addition will remain within the scope of the existing historical sources. revision: yes

Circularity Check

0 steps flagged

No circularity: purely historical narrative with no derivation chain

full rationale

The manuscript is a short historical note recounting the creation, transmission, and 1935 publication of Barrett's 1903 formula. It contains no equations, no fitted parameters, no self-citations used as load-bearing premises, and no claims that reduce to the paper's own inputs by construction. All substantive content depends on external historical sourcing rather than internal self-reference or renaming of results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a historical review paper containing no mathematical derivations, free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5581 in / 952 out tokens · 36641 ms · 2026-05-18T17:36:36.357596+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Paraguayan Sorrow: Writings of Rafael Barrett, Monthly Review Press

Barrett, R (2024). Paraguayan Sorrow: Writings of Rafael Barrett, Monthly Review Press

work page 2024
[2]

Mirando Vivir, (1912)

Barrett, R. Mirando Vivir, (1912). O.M. Bertani (editor) Montevideo (Kindle edition 2014)

work page 1912
[3]

García de Zúñiga, E. (193 5). Rafael Barrett, Matemático, Boletín de la Facultad de Ingeniería, 30-32 (https://www.edu.uy/es/node/39537)

work page
[4]

Massera, J. L. and Sheffer, J, J. (1966). Linear Differential Equations and Function Spaces, Academic Press, New York

work page 1966
[5]

Tenembaum, G and Mendès France, M. (2014). Les nombres premières, entre l'ordre et le chaos, Dunod, Paris. APPENDIX: Proof of Barrett's formula We take the terms of the summation, (k 1)!sin k sin k     (A1) and we analyze what happens if k is composite or if k is prime. Previously, let us remember the elementary properties of sin and cos: i...

work page 2014
[6]

absorbed

k is composite In this case, the prime factors of k are in (k 1)! . Here follows the argument that proves this assertion. Let iikp  . Legendre's formula allows us to determine the exponents of the prime factors of a factorial pi j j1 i k1v ((k 1)!) p      , (A2) where each term is the integer part ("floor" function) of the displayed quoti...

work page
[7]

We write it as follows: (k 1)! 1 1 [(k 1)! 1] 1 k k k      

k is prime In this case, we analyze the quotient (A3). We write it as follows: (k 1)! 1 1 [(k 1)! 1] 1 k k k       . (A4) But by Wilson's theorem, since k is prime, it is (k 1)! 1 0(mod k) k    (where k means that is a multiple of k) . Therefore, equation (A4) can be expressed as follows: 1od k (A5) with od an odd integer. Let us now evaluate ...

work page

[1] [1]

Paraguayan Sorrow: Writings of Rafael Barrett, Monthly Review Press

Barrett, R (2024). Paraguayan Sorrow: Writings of Rafael Barrett, Monthly Review Press

work page 2024

[2] [2]

Mirando Vivir, (1912)

Barrett, R. Mirando Vivir, (1912). O.M. Bertani (editor) Montevideo (Kindle edition 2014)

work page 1912

[3] [3]

García de Zúñiga, E. (193 5). Rafael Barrett, Matemático, Boletín de la Facultad de Ingeniería, 30-32 (https://www.edu.uy/es/node/39537)

work page

[4] [4]

Massera, J. L. and Sheffer, J, J. (1966). Linear Differential Equations and Function Spaces, Academic Press, New York

work page 1966

[5] [5]

Tenembaum, G and Mendès France, M. (2014). Les nombres premières, entre l'ordre et le chaos, Dunod, Paris. APPENDIX: Proof of Barrett's formula We take the terms of the summation, (k 1)!sin k sin k     (A1) and we analyze what happens if k is composite or if k is prime. Previously, let us remember the elementary properties of sin and cos: i...

work page 2014

[6] [6]

absorbed

k is composite In this case, the prime factors of k are in (k 1)! . Here follows the argument that proves this assertion. Let iikp  . Legendre's formula allows us to determine the exponents of the prime factors of a factorial pi j j1 i k1v ((k 1)!) p      , (A2) where each term is the integer part ("floor" function) of the displayed quoti...

work page

[7] [7]

We write it as follows: (k 1)! 1 1 [(k 1)! 1] 1 k k k      

k is prime In this case, we analyze the quotient (A3). We write it as follows: (k 1)! 1 1 [(k 1)! 1] 1 k k k       . (A4) But by Wilson's theorem, since k is prime, it is (k 1)! 1 0(mod k) k    (where k means that is a multiple of k) . Therefore, equation (A4) can be expressed as follows: 1od k (A5) with od an odd integer. Let us now evaluate ...

work page