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arxiv: 2604.25566 · v1 · submitted 2026-04-28 · 🧮 math.NT

Some results on naive transcendence in the ring of integers modulo infinitely large primes

Toshiki Matsusaka , Shin-ichiro Seki This is my paper

Pith reviewed 2026-05-07 15:10 UTC · model grok-4.3

classification 🧮 math.NT
keywords naive transcendencetranscendence resultsring of integers modulo primesABC conjectureirrationalitylogarithms
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0 comments X

The pith

In the ring of integers modulo infinitely large primes, various numbers are naively transcendental, with strengthened results and new examples provided.

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

The paper is trying to establish that in the ring A, transcendence results can be strengthened to the naive sense by removing assumptions from earlier work, and to provide new examples of naively transcendental numbers. A sympathetic reader would care because naive transcendence is a stronger property that automatically implies the Rosen version, offering more robust conclusions about these numbers in this non-standard ring. The paper also shows that under the ABC conjecture, numbers like log_A(2) are irrational even if full transcendence is not established.

Core claim

This paper presents various transcendence results in the ring A. It strengthens earlier results by removing some of their assumptions and, in some cases, upgrading them to statements of naive transcendence. Several examples of naive transcendental numbers not previously in the literature are presented. Irrationality of numbers such as log_A(2) is proven under the ABC conjecture.

What carries the argument

The ring A of integers modulo infinitely large primes with its two notions of transcendence, where naive transcendence implies Rosen-style finite algebraic transcendence.

If this is right

  • Earlier transcendence results hold without some assumptions and in the stronger naive form.
  • New examples of naively transcendental numbers are identified.
  • Irrationality of log_A(2) and similar numbers is established conditionally on the ABC conjecture.

Where Pith is reading between the lines

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

  • The ABC conjecture may enable irrationality proofs for additional constants in this ring.
  • These results could motivate searching for more examples of naive transcendence in non-archimedean settings.

Load-bearing premise

The definitions of the ring A and the two notions of transcendence are consistent and well-defined in the setup.

What would settle it

A concrete example of a number satisfying a polynomial equation with integer coefficients in the naive sense within ring A would contradict a transcendence claim.

read the original abstract

This paper presents various transcendence results in the ring of integers modulo infinitely large primes $\mathcal{A}$. In the ring $\mathcal{A}$, one can consider two notions of transcendence. One is based on the notion of finite algebraic numbers introduced by Rosen, while the other is transcendence in the naive sense. It is known that transcendence in the latter sense automatically implies transcendence in the former sense. In this paper, we strengthen results of Anzawa-Funakura and Luca-Zudilin by removing some of their assumptions and, in some cases, upgrading them to statements of naive transcendence. We also present several examples of naive transcendental numbers that do not seem to have appeared previously in the literature. Although we are not able to establish naive transcendence for certain numbers, we prove the irrationality of numbers such as $\log_{\mathcal{A}}(2)$ under the ABC conjecture.

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 / 3 minor

Summary. The paper studies transcendence in the ring A of integers modulo infinitely large primes, distinguishing Rosen-style finite algebraic transcendence from naive transcendence (with the latter implying the former). It strengthens results of Anzawa-Funakura and Luca-Zudilin by removing some assumptions and upgrading certain statements to naive transcendence, supplies new examples of naive transcendental elements not previously appearing in the literature, and establishes the irrationality of log_A(2) and related quantities under the ABC conjecture.

Significance. If the strengthenings and new examples hold, the work extends the scope of known transcendence results in this nonstandard ring without introducing new free parameters or ad-hoc axioms. The conditional irrationality statements under ABC provide a concrete link to a major open conjecture, while the unconditional parts offer incremental but explicit improvements over the cited prior papers. The provision of previously undocumented examples is a modest but verifiable contribution to the literature on naive transcendence.

minor comments (3)
  1. [Abstract] Abstract: the claim that certain results are upgraded to 'naive transcendence' is not accompanied by an explicit list of which prior theorems are affected or which assumptions are dropped; this should be stated with section references in the introduction for clarity.
  2. [Introduction] The setup of the ring A and the implication 'naive transcendence implies Rosen-style transcendence' is treated as known; a brief self-contained reminder or citation to the precise statement in Rosen's work would aid readers unfamiliar with the framework.
  3. [Section on conditional results] The ABC-conditional irrationality proof for log_A(2) is presented as a fallback when naive transcendence cannot be established; the manuscript should indicate whether this conditional result is new or follows directly from the cited Luca-Zudilin framework.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, including the strengthening of prior results, new examples of naive transcendental elements, and the conditional irrationality statements under the ABC conjecture. We appreciate the recommendation for minor revision and will incorporate improvements to presentation and clarity as needed.

Circularity Check

0 steps flagged

No significant circularity; results build on external citations

full rationale

The paper defines the ring A and the two transcendence notions (Rosen-style finite algebraic and naive) in its setup, stating as known that naive transcendence implies the Rosen version. It then strengthens results from the external papers of Anzawa-Funakura and Luca-Zudilin (different authors) by removing assumptions and upgrading some to naive transcendence, presents new examples, and proves an ABC-conditional irrationality for log_A(2). No derivation step reduces by construction to a fitted parameter, self-definition, or self-citation chain within the paper; the central claims remain independent of internal inputs and rely on cited external work plus the ABC conjecture. This is the standard non-circular case for a strengthening paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the existence and basic properties of the non-standard ring A together with the two transcendence definitions; no free parameters are introduced in the abstract, and the ABC conjecture is treated as an external hypothesis rather than an axiom of the paper.

axioms (2)
  • domain assumption The ring A of integers modulo infinitely large primes is well-defined and supports the two notions of transcendence (Rosen finite algebraic and naive).
    Invoked in the opening setup of the abstract as the ambient structure for all results.
  • domain assumption Naive transcendence implies transcendence in the Rosen sense.
    Stated explicitly as known background before the new results.

pith-pipeline@v0.9.0 · 5444 in / 1382 out tokens · 44896 ms · 2026-05-07T15:10:24.717395+00:00 · methodology

discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On the finite transcendence of Frobenius traces for abelian varieties over $\mathbb{Q}$

    math.NT 2026-05 unverdicted novelty 6.0

    Establishes finite transcendence of Frobenius traces for non-CM elliptic curves over Q and extends the result to some abelian varieties over Q.

  2. On the finite transcendence of Frobenius traces for abelian varieties over $\mathbb{Q}$

    math.NT 2026-05 unverdicted novelty 5.0

    Establishes transcendence of Frobenius traces for non-CM elliptic curves over Q and for several abelian varieties over Q.

  3. Notes on Algebraic Properties and Non-Standard Analysis of the Ring of Integers Modulo Infinitely Large Primes

    math.NT 2026-05 unverdicted novelty 5.0

    The paper extends transcendence criteria by Anzawa-Funakura and Matsusaka-Seki to apply non-standard analysis to the study of transcendence over the ring of integers modulo infinitely large primes.

Reference graph

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