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arxiv: 2605.13460 · v2 · pith:OWOIBHOUnew · submitted 2026-05-13 · 🧮 math.NT

Wieferich Primes and Monogenic Trinomials

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keywords Wieferich primesmonogenic polynomialstrinomialspower integral basisalgebraic integersnumber fieldsirreducibility
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The pith

The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if the prime p is not a Wieferich prime.

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

The paper proves an equivalence between the monogenicity of a specific family of trinomials and the avoidance of the Wieferich prime condition. For each prime p, the polynomial F_p(x) = x^{2p} + 2x^p + 2 defines a number field whose ring of integers has the power basis generated by a root precisely when 2^{p-1} is not congruent to 1 modulo p squared. This criterion directly ties the arithmetic of Wieferich primes to the integrality properties in these fields. A sympathetic reader would care because it offers an explicit criterion to determine when these polynomials generate the full ring of integers. The result classifies monogenicity for this family in terms of a well-studied prime property.

Core claim

A prime p is called a Wieferich prime if 2^{p-1} ≡ 1 (mod p^2). The paper shows that the monic polynomial F_p(x) := x^{2p} + 2x^p + 2 of degree 2p is monogenic—that is, irreducible over the rationals and such that the powers of a root form an integral basis—if and only if p is not a Wieferich prime.

What carries the argument

The monogenicity of the trinomial F_p(x), shown equivalent to the failure of the Wieferich congruence 2^{p-1} ≡ 1 mod p^2 via irreducibility and index arguments.

Load-bearing premise

The only arithmetic obstruction to monogenicity for this specific trinomial is the Wieferich congruence condition on p.

What would settle it

Explicit verification that F_p(x) fails to be monogenic for some non-Wieferich prime p, or succeeds for some Wieferich prime p, would settle the claim.

read the original abstract

A prime $p$ is called a Wieferich prime if $2^{p-1}\equiv 1 \pmod{p^2}$. A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we show that ${\mathcal F}_p(x):=x^{2p}+2x^{p}+2$ is monogenic if and only if $p$ is not a Wieferich prime.

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

1 major / 2 minor

Summary. The manuscript proves that the trinomial F_p(x) := x^{2p} + 2x^p + 2 is monogenic over Z if and only if the prime p is not a Wieferich prime to base 2. Irreducibility follows from the Eisenstein criterion at the prime 2. Monogenicity is equivalent to the index [O_K : Z[θ]] equaling 1, and the p-adic valuation of this index is shown to be positive precisely when 2^{p-1} ≡ 1 mod p^2 via Dedekind's criterion applied to the polynomial.

Significance. The result supplies an explicit, infinite family of trinomials whose monogenicity is controlled by a single, well-studied arithmetic condition (the Wieferich congruence). It thereby links a classical question in elementary number theory to the existence of power integral bases in number fields of degree 2p. The equivalence is parameter-free and yields falsifiable predictions for the (still unknown) list of Wieferich primes.

major comments (1)
  1. [§4] §4 (index computation): the argument that v_q(index) = 0 for all q ≠ p when p is not Wieferich must be made fully explicit. While the p-part is controlled by the Wieferich condition, the manuscript must separately verify that no other prime divides the index (e.g., via an explicit discriminant formula or a uniform bound on possible ramified primes). If this step is only sketched, the 'only if' direction is not yet load-bearing.
minor comments (2)
  1. [§3] The statement of Dedekind's criterion in §3 should include the precise local condition used for the different; the current wording leaves the translation from the congruence to the valuation implicit.
  2. Notation: the symbol F_p is used both for the polynomial and (implicitly) for the field; a distinct symbol for the number field would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to strengthen the exposition in §4. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (index computation): the argument that v_q(index) = 0 for all q ≠ p when p is not Wieferich must be made fully explicit. While the p-part is controlled by the Wieferich condition, the manuscript must separately verify that no other prime divides the index (e.g., via an explicit discriminant formula or a uniform bound on possible ramified primes). If this step is only sketched, the 'only if' direction is not yet load-bearing.

    Authors: We agree that the verification for q ≠ p must be made fully explicit rather than sketched. In the revised version we will add a self-contained argument (either by deriving an explicit formula for the discriminant of F_p(x) or by establishing a uniform bound on the primes that can ramify in the extension and then applying Dedekind’s criterion at those primes) showing that v_q([O_K : Z[θ]]) = 0 whenever q ≠ p and the Wieferich condition fails. This will render the “only if” direction fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence between independently defined monogenicity and Wieferich condition

full rationale

The paper defines monogenicity of F_p(x) via the standard criteria of irreducibility over Q and the power basis forming an integral basis for Z[theta]. Wieferich primes are defined via the independent arithmetic condition 2^{p-1} ≡ 1 mod p^2. The claimed iff statement is an equivalence between these two externally defined notions. No self-definitional reduction, fitted-parameter prediction, or load-bearing self-citation appears in the abstract or described derivation chain. Standard tools such as Eisenstein irreducibility and Dedekind criterion for the p-adic index contribution are expected to be applied without circularity. The skeptic concern addresses completeness of the index bound for q ≠ p but does not indicate any definitional collapse or self-referential step.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable from the provided text.

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