arXiv:2607.19337v2

On the $\mathcal{A}$ -transcendence of a Champernowne-type constant

Shin-ichiro Seki

math.NT11J8111A4111N0511C08

Paper metadata

Original submission: July 21, 2026
Latest revision: August 13, 2026 · version 2
Claimed MSC: 11J81, 11A41, 11N05, 11C08
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 5 pages, to appear in the Canadian Mathematical Bulletin

Abstract

Let $p_n$ denote the $n$ -th prime. We prove that for every nonzero polynomial $f(x)\in\mathbb{Z}[x]$ , there exist infinitely many positive integers $n$ such that $p_n\nmid f(n)$ .

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements2 reported findingsCorrect

The polynomial nondivisibility theorem and its stated consequence for the Champernowne-type element of $\mathcal A$ are correct.

Theorem 1.1Correct

Polynomial values fail divisibility by the corresponding prime infinitely often

Pages 1 and 3–4 · Theorem 1.1 and Section 2 · arXiv:2607.19337v2

Assume that $p_n\mid f(n)$ for only finitely many $n$ , and put $d=\deg f$ and $k=d(d+1)$ . Maynard's bounded-gap theorem gives one fixed tuple $0=h_0<h_1<\cdots<h_k$ for which $p_{n+j}=p_n+h_j$ for infinitely many $n$ . A nonzero integer vector solving the $d(d+1)$ homogeneous equations $\sum_{j=0}^k j^u h_j^v c_j=0$ produces a nonzero polynomial $\Phi(x,y)$ . For the selected $n$ , the quotient $\Phi(n,p_n)/\prod_i(p_n+h_i)$ is an integer. Expanding each $(p_n+h_j)^{-1}$ through degree $d-1$ cancels every main term by the displayed linear relations, while the remainder is $O(n^d/p_n^{d+1})=o(1)$ . The integer must therefore vanish, contradicting the nonzero asymptotic $\Phi(n,p_n)\sim a_{I,J}n^{I+J}(\log n)^J$ .

Maynard, Small gaps between primes, Theorem 1.1 ↗

Corollary 1.4Correct

The Champernowne-type element is outside the integral closure $\mathcal C_{\mathcal A}$

Page 3 · Corollary 1.4 and the preceding equivalence · arXiv:2607.19337v2

For every nonzero $f\in\mathbb Z[x]$ , Theorem 1.1 gives infinitely many components $p_n$ on which $f(n)\not\equiv0\pmod{p_n}$ . Thus no nonzero polynomial over $\mathbb Q$ annihilates $\boldsymbol\pi(p)=(n\bmod p_n)_{p_n}$ in the reduced product: clearing denominators changes only finitely many components. Since algebraicity over the field $\mathbb Q$ is equivalent to integrality over $\mathbb Q$ , this proves $\boldsymbol\pi(p)\notin\mathcal C_{\mathcal A}$ .

02Proofs2 reported findingsCorrect

The proof of the main theorem is correct and complete, including the bounded-prime-pattern reduction, the linear-algebra cancellation, and the final integer-versus- $o(1)$ contradiction.

Section 2Correct and complete

Cancellation-polynomial proof of Theorem 1.1

Pages 3–4 · Equations (2.1)–(2.4) · arXiv:2607.19337v2

There are exactly $d(d+1)$ homogeneous constraints and $d(d+1)+1$ unknowns, so a nonzero integer solution exists. Evaluating $\Phi(x,y)$ at each $y=-h_i$ proves that $\Phi$ is not identically zero. The lexicographically selected leading term gives the stated prime-number-theorem asymptotic. The finite geometric-series identity is exact, and writing $f(n+j)=\sum_{u=0}^d b_u(n)j^u$ puts every truncated term inside the imposed ranges $0\leq u\leq d$ and $0\leq v\leq d-1$ . Finally, $p_n\sim n\log n$ makes $O(n^d/p_n^{d+1})$ tend to zero.

Maynard inputCorrect and complete

The bounded-gap theorem supplies the required fixed prime pattern

Page 3 · first paragraph of Section 2 · arXiv:2607.19337v2

Maynard's theorem gives $\liminf_{n\to\infty}(p_{n+k}-p_n)<\infty$ for every fixed $k$ . The vectors $(p_{n+j}-p_n)_{j=0}^k$ therefore range over a finite set along infinitely many bounded-diameter occurrences, and the pigeonhole principle yields one fixed strictly increasing tuple $(h_j)_{j=0}^k$ exactly as required.

Maynard, Small gaps between primes, Theorem 1.1 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.19337v2
Authors listed: Shin-ichiro Seki
Audit date: August 18, 2026

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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. 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On the A\mathcal{A}A-transcendence of a Champernowne-type constant

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Abstract

Audit summary

Detailed mathematical audit

Polynomial values fail divisibility by the corresponding prime infinitely often

The Champernowne-type element is outside the integral closure CA\mathcal C_{\mathcal A}CA​

Cancellation-polynomial proof of Theorem 1.1

The bounded-gap theorem supplies the required fixed prime pattern

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On the $\mathcal{A}$ -transcendence of a Champernowne-type constant

The Champernowne-type element is outside the integral closure $\mathcal C_{\mathcal A}$