arXiv:2607.27774v2

Irrationality of finite logarithms in a congruence-class adèle ring

Daniel Evans

math.NT11J7211A4111N1311C08

Paper metadata

Original submission: July 30, 2026
Latest revision: August 13, 2026 · version 2
Claimed MSC: 11J72, 11A41, 11N13, 11C08
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: Version 2: 12 pages, included theorem 1.3 on squares of finite logarithms

Abstract

Finite logarithms can be defined in the ``poor man's adèle ring" $\mathcal{A}$ using Fermat quotients modulo sufficiently large primes. This ring contains $\mathbb{Q}$ and outside the trivial cases, Matsusaka and Seki have shown that finite logarithms cannot take non-zero rational values in $\mathcal{A}$ . Furthermore, a theorem of Silverman shows they are not zero, assuming the $abc$ -conjecture. We extend these results to primes restricted to arithmetic progressions of the form $p\equiv 1\bmod m$ by relating Fermat quotients to values of cyclotomic polynomials and their logarithmic derivatives. A signed version of the same argument shows unconditionally that the square of a finite logarithm cannot take non-zero rational values in $\mathcal{A}$ . As a further application, we show that, subject to the $abc$ -conjecture, finite logarithms cannot be quadratic irrational elements of $\mathcal{A}$ in Rosen's theory of finite algebraic numbers.

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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.

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Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The restricted irrationality theorem, the signed-square theorem, the classification of recurrence-degree-at-most-two elements, and the resulting quadratic obstruction are correct under their stated hypotheses. The two zero-value conclusions are explicitly conditional on the $abc$ -conjecture.

Theorem 1.2Correct

Finite logarithms on primes congruent to $1$ modulo $m$

Pages 2 and 5–8 · Theorem 1.2 and Sections 3.1–3.2 · arXiv:2607.27774v2

For $\alpha\in\mathbb Q^\times\setminus\{\pm1\}$ and $m\geq1$ , the cyclotomic construction with $W_\ell=\Phi_{m\ell}(u,v)$ proves unconditionally that $\log_{\mathcal A(m)}(\alpha)$ cannot be a nonzero rational element. Under the $abc$ -conjecture, the powerful-part estimate applied to $u^{m\ell}-v^{m\ell}$ rules out the zero element. The reductions from an arbitrary rational $\alpha$ to coprime integers $u>v\geq1$ preserve both claims.

Full paper, version 2 ↗

Theorem 1.3Correct

Squares of finite logarithms

Pages 2 and 8–9 · Theorem 1.3 and Section 3.3 · arXiv:2607.27774v2

If $\log_{\mathcal A}(\alpha)^2=c\in\mathbb Q^\times$ , then $c$ must be a rational square and the Fermat quotients equal one of two fixed signs modulo every sufficiently large prime. The signed cyclotomic sum deals with both signs: reduction modulo $\ell$ handles a nonzero sign sum, while reduction modulo the auxiliary prime $m$ handles a zero sign sum. This gives the unconditional nonzero-rational exclusion; reducedness of $\mathcal A$ and the conditional nonvanishing theorem give the $abc$ -conditional zero exclusion.

Full paper, version 2 ↗

Proposition 4.1 and Theorem 1.4Correct

Recurrence degree at most two and the quadratic obstruction

Pages 3 and 10–11 · Proposition 4.1 and Theorem 1.4 · arXiv:2607.27774v2

Every element of recurrence degree at most two is correctly reduced to $r+s\left(\frac{D}{p}\right)$ , with degree exactly two precisely when $s\neq0$ . Restricting to primes $p\equiv1\pmod N$ , where $N$ is the conductor of the quadratic character, forces $r+s=0$ by Theorem 1.2(i). Under the $abc$ -conjecture the same restriction would make the finite logarithm zero, contradicting Theorem 1.2(ii), and hence its recurrence degree is greater than two.

Full paper, version 2 ↗

02Proofs4 reported findingsCorrect

The proofs of the central statements and their material cyclotomic, $abc$ -conditional, and recurrence-theoretic inputs are correct and complete. The congruence conditions, exceptional primes, square-free or powerful alternatives, and all limiting estimates were checked in the exact reviewed version.

Lemmas 2.1–2.3 and Theorem 1.2(i)Correct and complete

Cyclotomic congruences and the nonzero-rational contradiction

Pages 3–6 · Lemmas 2.1–2.3 and Section 3.1 · arXiv:2607.27774v2

A prime $p\mid\Phi_n(u,v)$ with $p\nmid n$ has order $n$ for $u/v$ modulo $p$ , and Lemma 2.2 correctly obtains $\Phi_n(u,v)/p\equiv q_p(\alpha)v\Phi_{n,Y}(u,v)\pmod p$ . For $n=m\ell$ , every divisor of $W_\ell$ lies in the required congruence class and is simple under the rational-value assumption. Thus $W_\ell\mid T_\ell$ . Lemma 2.3 and the logarithmic-derivative estimate give $T_\ell/W_\ell\to0$ , while reduction modulo $\ell$ makes every quotient a nonzero integer, giving the contradiction.

Full paper, version 2 ↗

Lemma 3.1 and Theorem 1.2(ii)Correct and complete

The $abc$ -conditional powerful-part argument

Pages 7–8 · Lemma 3.1 and Section 3.2 · arXiv:2607.27774v2

The $abc$ -conjecture yields $B_n\ll_{\alpha,\varepsilon}u^{\varepsilon n}$ for the powerful part of $u^n-v^n$ . If the restricted finite logarithm vanished, Lemma 2.2 would make every prime divisor of $W_\ell$ occur at least twice, so $W_\ell$ would divide $B_{m\ell}$ . The resulting upper bound $W_\ell\ll u^{\ell\varphi(m)/2}$ contradicts the elementary cyclotomic lower bound $W_\ell\gg u^{\ell\varphi(m)}$ .

Silverman's $abc$-conditional non-Wieferich argument ↗

Theorem 1.3(i)Correct and complete

Signed cyclotomic construction

Pages 8–9 · Proof of Theorem 1.3 · arXiv:2607.27774v2

The signed sum $S_\ell^\varepsilon$ preserves divisibility by every prime factor of $W_\ell$ and has the same vanishing size bound as the unsigned sum. If $E_\ell=\sum_{p\mid W_\ell}\varepsilon_p$ is nonzero, its strict bound $|E_\ell|<\ell-1$ gives nonvanishing modulo $\ell$ . If $E_\ell=0$ , the choices $m\nmid av(u-v)$ and $\ell\equiv1\pmod{m(m-1)}$ give $R_\ell\equiv v(u-v)^{m-2}\not\equiv0\pmod m$ . The two cases exhaust the possibilities and establish the required nonzero-integer contradiction.

Full paper, version 2 ↗

Proposition 4.1 and Theorem 1.4Correct and complete

Quadratic recurrence classification and restriction to split primes

Pages 10–11 · Section 4 · arXiv:2607.27774v2

Reducible quadratic recurrences become constant modulo sufficiently large primes by Fermat's theorem. For an irreducible quadratic, Frobenius either fixes or exchanges the two roots, producing exactly the two values $c_+$ and $c_-$ and hence the quadratic-character form. The converse construction realizes every such form by a rational recurrence. The split-prime restriction in Theorem 1.4 then invokes Theorem 1.2 with matching modulus and hypotheses.

Rosen–Takeyama–Tasaka–Yamamoto, finite algebraic numbers ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.27774v2
Authors listed: Daniel Evans
Audit date: August 15, 2026

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Irrationality of finite logarithms in a congruence-class adèle ring

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Finite logarithms on primes congruent to 111 modulo mmm

Squares of finite logarithms

Recurrence degree at most two and the quadratic obstruction

Cyclotomic congruences and the nonzero-rational contradiction

The abcabcabc-conditional powerful-part argument

Signed cyclotomic construction

Quadratic recurrence classification and restriction to split primes

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Finite logarithms on primes congruent to $1$ modulo $m$

The $abc$ -conditional powerful-part argument