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arxiv: 2502.01345 · v2 · submitted 2025-02-03 · 🧮 math.NT

Sums of two units in number fields

Magdal\'ena Tinkov\'a , Robin Visser , Pavlo Yatsyna This is my paper

Pith reviewed 2026-05-23 04:32 UTC · model grok-4.3

classification 🧮 math.NT
keywords number fieldsunitsunit equationsfinitenesscubic fieldsDirichlet unit theoremsums of units
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The pith

In number fields without real quadratic subfields, only finitely many positive integers arise as sums of two units.

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

The paper defines N_K as the set of positive integers n that equal the sum of two units in the ring of integers of a number field K. It proves this set is finite whenever K contains no real quadratic subfield, using the rank and signature of the unit group. For cubic fields that are cyclic or have negative discriminant the authors list every solution explicitly. A reader cares because the result limits additive relations among units, which recur in Diophantine equations and class-number problems.

Core claim

Let K be a number field with ring of integers O_K. Let N_K be the set of positive integers n such that there exist units ε, δ in O_K^× satisfying ε + δ = n. We show that N_K is a finite set if K does not contain any real quadratic subfield. In the case where K is a cubic field, we also explicitly classify all solutions to the unit equation ε + δ = n when K is either cyclic or has negative discriminant.

What carries the argument

The set N_K of positive integers expressible as sums of two units from O_K^×.

If this is right

  • N_K is finite for all totally complex fields and for all fields whose only real embeddings come from a single real quadratic factor or none.
  • For cubic fields that are cyclic or have negative discriminant, every solution (ε, δ, n) can be written down in a finite list.
  • The presence or absence of a real quadratic subfield sharply separates fields with finite versus possibly infinite N_K.

Where Pith is reading between the lines

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

  • Fields that do contain a real quadratic subfield are expected to admit infinitely many such sums because units can grow without bound along the two real directions.
  • The finiteness result supplies an effective bound once the unit group is known, allowing in-principle computation of N_K for any concrete field satisfying the hypothesis.
  • The same distinction may govern other additive problems involving units, such as sums of three or more units.

Load-bearing premise

The unit group obeys the standard rank and signature given by Dirichlet's theorem, so that real quadratic subfields are the only case allowing unbounded sums.

What would settle it

An explicit number field with no real quadratic subfield together with an infinite list of distinct positive integers each equal to a sum of two of its units would disprove the claim.

read the original abstract

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{N}_K$ be the set of positive integers $n$ such that there exist units $\varepsilon, \delta \in \mathcal{O}_K^\times$ satisfying $\varepsilon + \delta = n$. We show that $\mathcal{N}_K$ is a finite set if $K$ does not contain any real quadratic subfield. In the case where $K$ is a cubic field, we also explicitly classify all solutions to the unit equation $\varepsilon + \delta = n$ when $K$ is either cyclic or has negative discriminant.

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 defines N_K as the set of positive integers expressible as sums of two units from O_K^x. It proves that N_K is finite whenever the number field K contains no real quadratic subfield. For cubic fields it further classifies all solutions to ε + δ = n when K is cyclic or has negative discriminant.

Significance. The result gives a clean, Dirichlet-unit-theorem-based criterion for boundedness of unit sums, distinguishing fields by the presence or absence of a rank-1 real unit group. The cubic classification supplies explicit, checkable data. If the proofs are complete and self-contained, the work strengthens the literature on unit equations and S-unit equations in number fields.

minor comments (3)
  1. The abstract states the main theorems but the manuscript should include a short outline of the proof strategy (e.g., reduction to the action of the unit group on the embedding space) already in the introduction for readability.
  2. In the cubic classification section, verify that all listed solutions satisfy the unit equation by direct computation and state the range of n explicitly rather than leaving it implicit.
  3. Add a reference to the classical results on the equation x + y = 1 in units (e.g., Siegel, Evertse) to situate the finiteness statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing its significance in the context of unit equations, and for recommending minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; standard application of Dirichlet unit theorem

full rationale

The paper states a finiteness result for the set N_K of positive integers expressible as sums of two units, conditioned on the absence of real quadratic subfields. This follows directly from the rank of the unit group O_K^x being r1 + r2 - 1 (Dirichlet's theorem), which is an external, independently established fact not derived within the paper. The abstract and claimed theorem contain no equations that redefine quantities in terms of themselves, no fitted parameters presented as predictions, and no load-bearing self-citations that would make the central claim tautological. The cubic-field classification is presented as an explicit enumeration, not a renaming or self-referential construction. The derivation chain is self-contained against external benchmarks and exhibits no reduction of outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the paper is expected to rest on standard algebraic number theory.

axioms (1)
  • standard math Dirichlet's unit theorem: the unit group O_K^× is finitely generated of rank r1 + r2 - 1
    Implicitly used to control the possible units whose sums can produce infinitely many n.

pith-pipeline@v0.9.0 · 5636 in / 1152 out tokens · 53526 ms · 2026-05-23T04:32:57.513168+00:00 · methodology

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Reference graph

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