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REVIEW 4 major objections 3 minor 6 references

Calculating the natural density of Mersenne numbers using nonstandard mathematical analysis

T0 review · 4 major / 3 minor · reviewed 2026-07-07 · glm-5.2

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Pith's one-line read Mersenne number density equals reciprocal of odd harmonic sum

desk verdict Circular column-density scaling: ε/(2x+1) is defined in terms of ε, making ε·ω=1 true by construction. read the letter →

arxiv 2607.05301 v1 pith:CURZ3IJ7 submitted 2026-07-06 math.NT math.CO

Gennady Eremin This is my paper · ORCID
classification math.NTmath.CO MSC 11B8311U1003H15
keywords MersennenumbersnaturaldensitynonstandardanalysisPepis-KalmarpairingfunctionThabitinfinitesimalsnumbertheorymatrixof
topics The Collatz Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper attempts to resolve the open question of the natural density of Mersenne numbers (numbers of the form 2^n − 1) by arranging all non-negative integers into a two-dimensional matrix via the Pepis-Kalmar pairing function f(x, y) = (2x+1)·2^y − 1. In this matrix, each column is a 'Mersenne tree': a sequence generated by repeatedly applying t ↦ 2t+1, starting from an even root. The first column contains the Mersenne numbers themselves; the second contains Thabit numbers (3·2^n − 1); the third contains 5·2^n − 1; and so on. The author argues that the natural density of column x is ε/(2x+1), where ε is the (infinitesimal) density of the Mersenne column. Summing all column densities to 1 (since the matrix partitions all of ℕ) yields ε·(1 + 1/3 + 1/5 + 1/7 + ...) = 1, so ε = 1/ω, where ω is the sum of reciprocals of odd natural numbers. The author treats this as an equation in nonstandard analysis, where ε is a genuine infinitesimal and ω is a genuinely infinite hyperreal, concluding that the Mersenne density is positive (nonzero) and equal to the reciprocal of the odd harmonic sum.

What carries the argument

The Pepis-Kalmar pairing function f(x, y) = (2x+1)·2^y − 1 organizes all non-negative integers into a matrix whose columns are disjoint 'Mersenne trees'. Row densities form a geometric series summing to 1. Column densities are assumed to scale as ε/(2x+1), and the requirement that column densities also sum to 1 produces the equation ε·ω = 1.

What would settle it

If the column densities do not scale as ε/(2x+1), then the equation ε·ω = 1 does not follow. A standard-analysis check would ask whether the partial sums of column counting functions are consistent with this scaling in the limit.

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Extended reading notes

Core claim

The central claim is that the natural density ε of Mersenne numbers satisfies ε × ω = 1, where ω = 1 + 1/3 + 1/5 + 1/7 + ... is the sum of reciprocals of odd natural numbers. This is derived by assigning density ε/(2x+1) to column x of a pairing-function matrix and requiring that all column densities sum to 1. The result is presented within the framework of nonstandard analysis, where ε and 1/ω are equivalent infinitesimals, and the conclusion is that the Mersenne density is strictly positive.

Load-bearing premise

The argument depends on the claim that the density of column x equals ε/(2x+1), where ε is the Mersenne column's density. This scaling is justified by observing that the range needed to contain n+1 terms of column x has length (2x+1)·2^n, which is (2x+1) times the corresponding range for Mersenne numbers. But this reasoning defines each column's density in terms of ε itself rather than deriving ε from an independent principle, so the equation ε·ω = 1 may be true by the setup,

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The manuscript proposes to compute the natural density of Mersenne numbers using nonstandard analysis. Natural numbers are arranged in a matrix via the Pepis–Kalmar pairing function f(x,y) = (2x+1)·2^y − 1, so that column x=0 contains Mersenne numbers, column x=1 contains Thabit numbers, etc. The author argues that the density of column x is ε/(2x+1), where ε is the (infinitesimal) density of Mersenne numbers. Summing over all columns yields ε·(1 + 1/3 + 1/5 + ···) = ε·ω = 1, giving ε = 1/ω (Theorem 2). The paper claims this resolves an open problem.

Significance. The question of densities of sparse sequences in a nonstandard framework could be of interest if treated rigorously. However, the central result does not hold up under scrutiny (see below).

major comments (4)
  1. §3, the column-density scaling: The claim that the natural density of column x equals ε/(2x+1) is based on comparing counting functions at different cutoff points. For Mersenne numbers, the author counts n+1 values in [0, 2^n−1] (density (n+1)/2^n). For Thabit numbers, the author counts n+1 values in [0, 3·2^n−1] (density (n+1)/(3·2^n)). The ratio 1/3 is obtained by comparing densities at different ranges. Natural density requires a common cutoff N for all sequences. With a common cutoff N, the number of Mersenne numbers up to N is ~log₂(N)+1, and the number of Thabit numbers up to N is ~log₂(N/3)+1 = log₂(N)−log₂(3)+1. Both densities are ~log₂(N)/N → 0, and their ratio tends to 1, not 1/3. The scaling ε/(2x+1) is therefore incorrect for natural density, and the entire derivation of Eq. (2) collapses.
  2. §3 and §4, circularity of Eq. (2): Even setting aside the incorrect scaling, the derivation is circular. The column densities are defined as ε/(2x+1) in terms of ε itself (§3: 'it is logical to accept that the natural density of Thabit numbers is equal to ε/3'). Summing these defined quantities gives ε·ω = 1 by construction; it does not independently determine ε. The conclusion ε = 1/ω is forced by the definition of the column densities, not derived from independent principles.
  3. §1 and §3, the 'open problem' framing: The natural density of Mersenne numbers {2^n−1} is not an open problem. The counting function is ⌊log₂(N+1)⌋+1 = O(log N), so the natural density is lim_{N→∞} O(log N)/N = 0. The author acknowledges this ('the density of Mersenne numbers tends to zero') but then attempts to recover a nonzero infinitesimal density via informal NSA arguments. No transfer principle, hyperfinite counting argument, or Loeb measure construction is provided that would justify treating the standard-part-zero density as a nonzero infinitesimal in a way that contradicts the standard result.
  4. §4, Theorem 2: The theorem states ε = 1/ω where ω = 1 + 1/3 + 1/5 + ···. In standard analysis this series diverges; in NSA, ω depends on the choice of hyperinteger cutoff and is not uniquely defined. The paper does not specify which hyperfinite partial sum ω refers to, so 1/ω is not a well-defined infinitesimal. Without a precise definition of ω, Theorem 2 has no definite mathematical content.
minor comments (3)
  1. The notation switches between ε and ɛ; consistency would help readability.
  2. Reference [5] is dated 2026 and cites Wikipedia; a textbook reference for natural density would be more appropriate.
  3. Proposition 1 is stated and proved, but Theorem 1 is referenced nowhere in the text; only Theorem 2 appears. This numbering gap should be resolved.

Simulated Author's Rebuttal

4 responses · 3 unresolved

We thank the referee for a careful and substantive report. The referee raises four major objections concerning the column-density scaling, circularity of the derivation, the 'open problem' framing, and the well-definedness of ω. After careful consideration, we acknowledge that the referee's points on the scaling argument and circularity are correct and cannot be adequately answered in the current framework. We also concede the 'open problem' framing is inaccurate. We offer a partial defense on the NSA treatment of ω but acknowledge the paper requires substantial revision. Given the severity of the issues identified, we agree the manuscript in its current form does not meet the standard for publication.

read point-by-point responses

  1. Referee: §3, the column-density scaling: The claim that the natural density of column x equals ε/(2x+1) is based on comparing counting functions at different cutoff points. With a common cutoff N, both Mersenne and Thabit densities are ~log₂(N)/N → 0, and their ratio tends to 1, not 1/3. The scaling ε/(2x+1) is incorrect for natural density.

    Authors: The referee is correct. The derivation in §3 compares densities at different cutoffs: (n+1)/2^n for Mersenne numbers versus (n+1)/(3·2^n) for Thabit numbers. These ranges have different lengths (2^n versus 3·2^n), so the ratio 1/3 is an artifact of comparing densities at different scales. Under a common cutoff N, both counting functions are O(log N), both densities are O(log N / N) → 0, and their ratio tends to 1. The scaling ε/(2x+1) does not hold for natural density. We cannot defend this argument and concede that Eq. (2) collapses without it. revision: yes

  2. Referee: §3 and §4, circularity of Eq. (2): The column densities are defined as ε/(2x+1) in terms of ε itself. Summing these defined quantities gives ε·ω = 1 by construction; it does not independently determine ε.

    Authors: The referee is correct. The column densities are defined in terms of ε (e.g., 'it is logical to accept that the natural density of Thabit numbers is equal to ε/3'), and summing these defined quantities yields ε·ω = 1 by construction. This is circular: ε is not determined by an independent argument but is simply posited and then 'recovered' from the sum. We acknowledge this circularity and cannot resolve it within the paper's current framework. revision: yes

  3. Referee: §1 and §3, the 'open problem' framing: The natural density of Mersenne numbers is not open. The counting function is O(log N), so the density is 0. No transfer principle, hyperfinite counting argument, or Loeb measure construction is provided to justify treating this as a nonzero infinitesimal.

    Authors: The referee is correct that the standard natural density of Mersenne numbers is zero and that this is not an open problem. The counting function ⌊log₂(N+1)⌋+1 = O(log N) gives density lim O(log N)/N = 0. The abstract and §1 incorrectly frame this as an unsolved problem. We will correct this framing. Regarding the NSA treatment: the paper does not invoke the transfer principle, hyperfinite counting, or Loeb measures. The informal NSA arguments presented do not justify treating the standard-part-zero density as a well-defined nonzero infinitesimal. We acknowledge this gap and cannot defend the claim that the paper provides a rigorous NSA derivation. revision: yes

  4. Referee: §4, Theorem 2: ω = 1 + 1/3 + 1/5 + ... diverges in standard analysis; in NSA, ω depends on the choice of hyperinteger cutoff and is not uniquely defined. The paper does not specify which hyperfinite partial sum ω refers to, so 1/ω is not well-defined.

    Authors: We partially agree. The referee is correct that ω is not uniquely defined without specifying a hyperinteger cutoff, and the paper does not do so. This is a genuine gap. However, we note that in NSA, specifying a particular hyperinteger H and taking ω_H = Σ_{k=0}^{H} 1/(2k+1) would yield a specific infinitesimal 1/ω_H. The value would depend on H, but different choices of H (in the same galaxy) would yield infinitesimals of the same order. That said, since the scaling argument underlying Eq. (2) is incorrect (as acknowledged above), the well-definedness of ω is moot for the paper's main result. We concede the point. revision: partial

standing simulated objections not resolved
  • The column-density scaling ε/(2x+1) is incorrect for natural density, as the referee demonstrates with the common-cutoff argument. This is a fatal flaw that we cannot overcome.
  • The derivation of ε·ω = 1 is circular, as the column densities are defined in terms of ε itself. We cannot provide a non-circular derivation within the current framework.
  • The paper does not provide the rigorous NSA machinery (transfer principle, hyperfinite counting, Loeb measures) needed to justify treating the zero standard density as a nonzero infinitesimal.

Circularity Check

2 steps flagged · score 9.0 of 10

Column densities defined as ε/(2x+1) in terms of ε itself; summing them yields ε·ω=1 by construction.

  1. self definitional [§3, column density definitions and §4 equation (2)]
    "it is logical to accept that the natural density of Thabit numbers is equal to ɛ/3 (we reduce the density of Mersenne numbers by a factor of three). ... we obtain another infinitesimal density equal to ɛ/5. ... 1 = ɛ + ɛ/3 + ɛ/5 + ɛ/7 + ... = ɛ × (1 + 1/3 + 1/5 + 1/7 + ...) = ɛ × ω."

    The paper defines the density of column x as ε/(2x+1), where ε is the Mersenne density being sought. This definition is stated as 'logical to accept' with no independent derivation. When these defined quantities are summed in equation (2), the result is ε·(1 + 1/3 + 1/5 + ...) = ε·ω = 1, hence ε = 1/ω. But this equation is forced entirely by the definition: the column densities were constructed as ε divided by odd numbers, so their sum is ε times the sum of reciprocals of odd numbers by construction. The conclusion ε = 1/ω is not derived from independent principles—it is algebraically equivalent to the ansatz used to define the column densities. The paper's central result (Theorem 2) is thus a restatement of its own definitional assumption.

  2. self citation load bearing [§4, final paragraph; reference [2]]
    "In the paper [2] we proved that the density of Mersenne numbers is positive (non-zero). Now we have obtained a concrete equation for determining the natural density of a sequence of Mersenne numbers."

    The paper cites [2] (Eremin, same author) as the proof that ε > 0, which is the load-bearing claim that distinguishes the result from the standard-analysis answer of 0. Without this self-cited result, the paper acknowledges ε could be 0, which would make every term ε/(2x+1) = 0 and the sum 0, not 1. The positivity of ε is not independently established within this paper; it depends entirely on the author's own prior work [2], which is not externally verified or machine-checked. This self-citation carries the central claim.

full rationale

The paper's derivation is circular at its core. In §3, the density of each matrix column is defined as ε/(2x+1), where ε is the very quantity the paper seeks to determine. The justification is that column x has (2x+1)·2^n range versus 2^n for Mersenne numbers, so the density is 'reduced by a factor of (2x+1).' But this scaling assumes ε is a nonzero infinitesimal that can be divided by odd numbers—a claim that is the paper's conclusion, not its premise. When these defined column densities are summed in equation (2), the result ε·ω = 1 follows algebraically from the definitions, not from any independent counting argument or transfer principle. The paper provides no hyperfinite counting argument, no transfer principle application, and no external verification that would make ε a well-defined nonzero infinitesimal independent of the conclusion. Additionally, the positivity of ε (the key claim distinguishing this from the standard result of 0) is supported only by self-citation [2]. The derivation chain reduces to: define column densities in terms of ε → sum them → recover ε = 1/ω by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The axiom ledger reveals the paper's structure: the key assumption (column density = ε/(2x+1)) is ad hoc and makes the final equation circular. The claim that the density is an open problem is incorrect. The nonstandard analysis framework is invoked without rigorous justification.

free parameters (2)
  • ε (Mersenne number density) = 1/ω (infinitesimal)
    ε is introduced as the infinitesimal density of Mersenne numbers. Its value is determined by the circular equation ε·ω = 1, which is true by construction given the column density assumption.
  • ω (sum of reciprocals of odd numbers) = infinitely large
    ω = 1 + 1/3 + 1/5 + ... is treated as an infinitely large number in nonstandard analysis. This is a standard divergent series; its treatment as a single hyperreal is a modeling choice.
assumptions (4)
  • ad hoc to paper The natural density of column x (containing (2x+1)·2^y - 1) equals ε/(2x+1).
    Stated in §3: 'it is logical to accept that the natural density of Thabit numbers is equal to ε/3.' This scaling is not derived from any independent principle and is the load-bearing assumption that makes the final equation circular.
  • domain assumption The total density of all columns equals 1.
    Stated in §4: 'the total natural density of all columns... is equal to the density of the set of natural numbers ℕ.' This is reasonable if the matrix partitions ℕ, but combined with the column density assumption it produces the circular result.
  • ad hoc to paper Nonstandard analysis infinitesimals can represent natural densities.
    The paper treats the limit (n+1)/2^n → 0 as a positive infinitesimal ε rather than zero. Natural density is a standard-analysis concept; the nonstandard framework is invoked without rigorous justification of how infinitesimals relate to the standard density.
  • ad hoc to paper The natural density of Mersenne numbers is an open problem.
    Stated in §1 and the abstract. The density is known to be zero by the standard argument that (n+1)/2^n → 0. The paper does not cite or engage with this standard result.
invented entities (1)
  • ε as a positive infinitesimal density of Mersenne numbers
    purpose: To represent the natural density of Mersenne numbers as a non-zero quantity in nonstandard analysis.
    No independent evidence is provided that the natural density is positive. The standard density is zero. The nonstandard framework does not change the standard density value.

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Cite this review

Pith. "Pith review of Calculating the natural density of Mersenne numbers using nonstandard mathematical analysis." pith.science (2026). https://pith.science/paper/CURZ3IJ7

@misc{pith2026260705301,
  author       = {Pith},
  title        = {Pith review of: Calculating the natural density of Mersenne numbers using nonstandard mathematical analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CURZ3IJ7}},
  note         = {Machine review of arXiv:2607.05301}
}
abstract

Currently, among the open (unsolved) problems in number theory is the following: it is unknown what is the natural density of the sequence of Mersenne numbers in the set of natural numbers. In the paper, using methods of nonstandard mathematical analysis, we obtain the following equation: the natural density of Mersenne numbers (some infinitesimal value $e$) multiplied by the sum of the reciprocals of odd numbers (the infinitely large value {\omega} = 1 + 1/3 + 1/5 + ...) is equal to 1, or the equality $e$ = 1/{\omega} is true. In nonstandard analysis, the resulting infinitesimal numbers $e$ and 1/{\omega} are considered equivalent. We obtained this result by working with a two-dimensional matrix of non-negative integers, where odd numbers are separated from even ones by the Pepis-Kalmar pairing function.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

  1. [1]

    M. Davis. Applied Nonstandard Analysis . New York: John Wiley & Sons, 1977

    work page 1977

  2. [2]

    G. Eremin. Infinite matrix of odd natural numbers. A bit about Sophie Germain primes, 2025. https://arxiv.org/abs/2501.17090

    work page Pith review arXiv 2025

  3. [3]

    Robinson

    A. Robinson. Non-Standard Analysis. Kon. Nederl. Akad. Wetensch. Amsterdam Proc. AM (=Indag. Math. 23), 1961, pp. 432-440

    work page 1961

  4. [4]

    V. A. Uspensky, What is nonstandard analysis ? Nauka Publishers, M., 1987, 128 pp. (Russian)

    work page 1987

  5. [5]

    Natural density, 2026

    Wikipedia. Natural density, 2026. https://en.wikipedia.org/wiki/Natural_density

    work page 2026

  6. [6]

    N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences , OEIS Foundation Inc., 2026. https://oeis.org/ Email address: ergenns@gmail.com Written: July 7, 2026

    work page 2026

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