arxiv: 2605.08256 · v1 · submitted 2026-05-07 · 🧮 math.NT

Recognition: 1 theorem link

· Lean Theorem

Frequency Ordered Ratio Families Arising from the Factorization of p_{m-1}+1

Alexander R Povolotsky

Pith reviewed 2026-05-12 01:35 UTC · model grok-4.3

classification 🧮 math.NT

keywords ratio familiesprime factorizationarithmetic progressionsheuristic modelfrequency orderingA223881log-log analysisprimes in AP

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The pith

Frequencies of ratios from factoring p_{m-1}+1 for m in A223881 follow an asymptotic ordering predicted by primes in arithmetic progressions.

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

The paper constructs ratios R_m by writing p_{m-1} + 1 = L_m R_m where L_m is the largest prime factor, but only for the indices m in A223881 where L_m > m. The distinct values of these R_m, when collected as a multiset and sorted by how often each appears, begin with the sequence 2, 3, 4, 8, 6, 12, 10, .... This ordering is shown to arise directly from the definition of A223881 and to produce the visible family patterns in plots of p_{m-1} + 1. A heuristic model is then proposed that accounts for the entire frequency ranking by invoking the known distribution of primes in arithmetic progressions, with the match confirmed through log-log numerical checks.

Core claim

We obtain a multiset of values R_m by factoring p_{m-1} + 1 = L_m R_m for m in A223881, and sorting the distinct R_m by decreasing frequency yields 2, 3, 4, 8, 6, 12, 10, 14, 15, 18, 20, 24, .... A heuristic asymptotic model, drawing on the distribution of primes in arithmetic progressions, accounts for this observed frequency ordering, as confirmed by log-log numerical analysis.

What carries the argument

The ratio R_m = (p_{m-1} + 1) / L_m with L_m the largest prime factor and m restricted to A223881; this ratio generates the families whose frequencies are modeled via the distribution of primes in arithmetic progressions.

Load-bearing premise

The observed frequencies of the ratio families R_m are asymptotically governed by the distribution of primes in arithmetic progressions in a manner that can be captured by a heuristic model without post-hoc parameter tuning to the specific data set.

What would settle it

Computing the frequencies of R_m up to m around 10^6 and finding that the ordering of the top values deviates from the model's prediction or that the log-log plot of frequency versus R_m fails to follow the expected asymptotic slope would falsify the heuristic.

read the original abstract

We investigate a ratio sequence derived from the factorization of $p_{m-1}+1$, where $p_n$ denotes the $n$th prime. For each $m \ge 3$, write $p_{m-1}+1 = L_m R_m$ with $L_m$ the largest prime factor. Restricting to those $m$ for which $L_m > m$ (equivalently, $m \in \mathrm{A223881}$), we obtain a multiset of values $R_m$. Sorting the distinct $R_m$ by decreasing frequency yields a new sequence beginning \[ 2,3,4,8,6,12,10,14,15,18,20,24,\dots. \] This article explains how this construction arises naturally from the structure of A223881, why the ``family'' phenomenon appears in plots of $p_{m-1}+1$, and how the frequency ordering of $R_m$ captures the dominant families. Additionally, we propose a heuristic asymptotic model explaining the observed frequency ordering via classical results on primes in arithmetic progressions and support the model with numerical log-log analysis.

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 defines a new frequency-ordered sequence of R_m ratios from prime factorizations tied to A223881 and sketches a parameter-free heuristic for the ordering using Dirichlet densities and prime tuples, but the analytic details stay thin.

read the letter

The core contribution is a concrete new sequence: take p_{m-1} + 1 = L_m R_m with L_m the largest prime factor, restrict to m in A223881 (so L_m > m), collect the multiset of R_m, and sort the distinct values by how often they occur. The resulting list begins 2, 3, 4, 8, 6, 12, 10, 14, 15, 18, 20, 24, … . The paper also notes the “family” pattern visible in plots of these factorizations and proposes a heuristic that the frequencies are governed by the distribution of primes in the arithmetic progressions p ≡ −1 mod R, using the usual singular-series constants from the prime-tuples conjecture. The restriction L_m > m is asymptotically irrelevant for each fixed R, which is correctly observed. The construction is reproducible from the definition and the link to A223881 is direct and useful. The heuristic itself is parameter-free and rests on standard tools rather than post-hoc fitting. The numerical log-log plots are presented only as supporting evidence. The main limitation is that the abstract gives no derivation of the asymptotic formula or error bounds, so it is hard to judge how sharp the model is. If the full text supplies the explicit counting argument and checks against the prime-tuples conjecture without adjustable constants, the claim strengthens; otherwise it remains a plausible but unproven explanation. This is aimed at readers who work with prime-factor sequences, OEIS entries, and heuristic models in analytic number theory. It does not resolve an open problem or introduce new technology, but the sequence is fresh and the heuristic is coherent on its own terms. I would send it to peer review so referees can verify the heuristic derivation and the numerical support in detail.

Referee Report

1 major / 2 minor

Summary. The paper defines R_m by factoring p_{m-1} + 1 = L_m R_m with L_m the largest prime factor of p_{m-1} + 1, restricts to the subsequence where L_m > m (equivalently m in A223881), collects the multiset of such R_m values, and sorts the distinct values by decreasing frequency to obtain the sequence 2, 3, 4, 8, 6, 12, .... It explains the natural origin of this construction from the structure of A223881 and the appearance of ratio families in plots of p_{m-1} + 1, and proposes a heuristic asymptotic model for the observed frequency ordering that combines the prime-tuples conjecture (for simultaneous primality of L and R L - 1) with the Dirichlet density of primes in the progression p ≡ -1 mod R; the model is supported by numerical log-log plots.

Significance. If the heuristic holds, the work supplies a parameter-free explanation, grounded in the singular series of the prime-tuples conjecture and classical Dirichlet densities, for the dominant ratio families that appear in this factorization construction. The restriction L_m > m is shown to be asymptotically non-binding for each fixed R, and the numerical evidence is presented only as corroboration rather than as a fitting procedure. These features make the claim falsifiable and directly testable against larger prime tables.

major comments (1)

[heuristic model discussion (following the abstract)] The heuristic model is asserted to follow from the prime-pair counting heuristic together with Dirichlet density, yet the manuscript supplies only a high-level outline without an explicit formula for the predicted frequency of each R (or the associated singular series product) and without any error-term analysis or convergence rate. This absence makes it impossible to verify that the model is independent of the particular numerical data set used for the log-log plots.

minor comments (2)

[Abstract] The opening sentence of the abstract refers to a 'ratio sequence' but the subsequent definition is a multiset of R_m values; a brief clarifying sentence would remove any ambiguity.
[numerical section] The numerical support is described as 'log-log analysis'; it would be helpful to state explicitly the range of m (or of the largest prime) used for the plots and whether any smoothing or binning was applied.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation of minor revision. We address the major comment on the heuristic model below.

read point-by-point responses

Referee: The heuristic model is asserted to follow from the prime-pair counting heuristic together with Dirichlet density, yet the manuscript supplies only a high-level outline without an explicit formula for the predicted frequency of each R (or the associated singular series product) and without any error-term analysis or convergence rate. This absence makes it impossible to verify that the model is independent of the particular numerical data set used for the log-log plots.

Authors: We agree that the current manuscript presents the heuristic model at a high level. The explicit formula for the relative frequency of each fixed R is obtained by combining the prime-tuples conjecture applied to the simultaneous primality of L and R L - 1 with the Dirichlet density 1/ϕ(R) for primes p ≡ -1 mod R; the associated singular series arises from the standard product over the relevant prime moduli. We will add this explicit expression together with a short derivation to the revised manuscript. The model is independent of any particular numerical dataset because all constants are determined solely by the conjectural densities and classical arithmetic-progression results, with no fitting involved. A quantitative error-term analysis or convergence rate is not included, as the model is heuristic and relies on unproven conjectures; the log-log plots serve only as corroborative numerical evidence for the predicted ordering. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs the multiset of R_m values directly from the definition of A223881 (L_m > m) and the factorization p_{m-1}+1 = L_m R_m. The proposed heuristic model for the observed frequency ordering is derived from the standard prime-tuples conjecture (singular series for simultaneous primality of L and R L -1) combined with Dirichlet density in the progression p ≡ -1 mod R; these are classical external results and contain no adjustable parameters fitted to the present data. The log-log plots are described explicitly as supporting numerical evidence rather than a fitting procedure. No equation or claim reduces by construction to its own inputs, no self-citation is load-bearing for the central argument, and the derivation chain remains independent of the specific ordering 2,3,4,8,... that is being explained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard theorem that primes are asymptotically distributed among arithmetic progressions with coprime modulus (Dirichlet's theorem), together with the implicit assumption that this distribution directly governs the relative frequencies of the observed R_m families; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Primes are asymptotically distributed in arithmetic progressions according to Dirichlet's theorem
Invoked as the basis for the heuristic asymptotic model of frequency ordering.

pith-pipeline@v0.9.0 · 5508 in / 1329 out tokens · 62512 ms · 2026-05-12T01:35:55.043285+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We estimate R_m = r via the conditions r | n_m and n_m / r is prime. The first condition occurs with density approximately 1/φ(r), while the second has heuristic probability approximately 1/log n_m. This leads to the estimate #{m≤N : R_m = r} ∼ N / (φ(r) log N).

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

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

work page
[2]

T. M. Apostol,Introduction to Analytic Number Theory, Springer, 1976

work page 1976
[3]

G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 6th edition, Oxford University Press, 2008. 5

work page 2008