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arxiv: 2310.14828 · v2 · pith:AKKGYG4Ynew · submitted 2023-10-23 · 🧮 math.PR

Divisibility and primality in random walks

Michel J. G. Weber This is my paper

Pith reviewed 2026-05-24 06:17 UTC · model grok-4.3

classification 🧮 math.PR
keywords random walksdivisibilityprimalityBernoulli random walkRademacher random walkCramér modelSelberg estimates
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The pith

Divisibility results for the Bernoulli random walk extend to wide classes of iid and independent non-iid walks, with new primality statements for the Rademacher walk and comparable divisor estimates in the Cramér model.

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

The paper aims to generalize divisibility results known for the Bernoulli random walk to much wider families of random walks that may have independent but non-identical steps. It also establishes new results on the primality of the walk positions in the Rademacher case and derives estimates for how the divisors of the walk values are distributed when the walk is viewed through the Cramér random model for primes. These extensions rely on independence assumptions and comparison estimates similar to those of Selberg. A sympathetic reader would care because the findings indicate that certain arithmetic features of random walks on the integers remain stable when the step distribution changes, as long as independence across time is kept.

Core claim

We improve or extend some of our divisibility results to wide classes of iid or independent non iid random walks. We also obtain new primality results for the Rademacher random walk. We study the value distribution of divisors of the random walk in the Cramér model, and obtain a general estimate of a similar kind to that of the Bernouilli model. Earlier results on divisors and quasi-prime numbers in the Bernoulli model are recorded, as well as some other recent for the Cramér random model, based on an estimate due to Selberg.

What carries the argument

Independence conditions on the increments of the random walk, together with the Cramér probabilistic model and Selberg-type estimates applied to divisor and primality counts.

If this is right

  • Divisibility by small integers occurs with positive density that can be estimated uniformly across the extended classes of walks.
  • Quasi-prime values of the walk appear with frequencies controlled by the same type of estimates used in the Bernoulli setting.
  • The distribution of divisors of walk positions in the Cramér model follows a general form parallel to the Bernoulli model.
  • Primality statements for the Rademacher walk give explicit conditions under which walk positions are prime.

Where Pith is reading between the lines

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

  • If the extensions hold, divisibility counts might be used to classify or compare the arithmetic behavior of different random walk models on the integers.
  • The same independence-based arguments could be tested on walks with steps drawn from continuous distributions to see whether similar divisor patterns persist.
  • Connections between these random-walk results and classical questions on prime factors in integer sequences generated by dependent increments become worth exploring numerically.

Load-bearing premise

The random walk steps satisfy independence (iid or independent non-iid) so that the divisibility and distribution results transfer from the Bernoulli case.

What would settle it

A concrete counterexample would be an independent non-iid random walk for which the density of positions divisible by a small fixed integer, such as 3, deviates from the density predicted by the Bernoulli analysis.

read the original abstract

In this paper we study the divisibility and primality properties of the Bernoulli random walk. We improve or extend some of our divisibility results to wide classes of iid or independent non iid random walks. We also obtain new primality results for the Rademacher random walk. We study the value distribution of divisors of the random walk in the Cram\'er model, and obtain a general estimate of a similar kind to that of the Bernouilli model. Earlier results on divisors and quasi-prime numbers in the Bernoulli model are recorded, as well as some other recent for the Cram\'er random model, based on an estimate due to Selberg.

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.

Referee Report

0 major / 1 minor

Summary. The manuscript examines divisibility and primality properties of the Bernoulli random walk. It extends or improves prior divisibility results to wide classes of iid and independent non-iid random walks, derives new primality results for the Rademacher random walk, and studies the value distribution of divisors in the Cramér model to obtain a general estimate paralleling the Bernoulli case (via Selberg estimates). Earlier results on divisors and quasi-primes in the Bernoulli model are recorded.

Significance. If the derivations hold, the work broadens the scope of divisibility and primality analysis for random walks under explicit independence hypotheses and supplies a model-comparison estimate that may be useful in probabilistic number theory.

minor comments (1)
  1. [Abstract] Abstract: 'Bernouilli' is misspelled; correct to 'Bernoulli'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends the author's prior divisibility results to broader classes of random walks under explicit iid or independent non-iid conditions and derives new primality statements for the Rademacher walk. The Cramér-model divisor estimate is presented as analogous to the Bernoulli case via an external Selberg estimate, with earlier Bernoulli-model results recorded only as background. No derivation step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the central claims rest on the stated independence hypotheses and model transfer rather than on any internal renaming or ansatz smuggling. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; work relies on standard definitions of Bernoulli/Rademacher walks and the known Cramér random model.

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