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arxiv: 2604.23404 · v2 · submitted 2026-04-25 · 🧮 math.NT

Differences of squares of upper-triangular 2times 2 integer matrices

Andrej Dujella , Zrinka Franu\v{s}i\'c This is my paper

Pith reviewed 2026-05-12 00:51 UTC · model grok-4.3

classification 🧮 math.NT
keywords upper-triangular matricesdifferences of squares2x2 integer matricesrepresentation criteriadivisibility conditionscongruence conditionsmatrix equationsnumber theory
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The pith

An upper-triangular 2x2 integer matrix can be written as the difference of squares of two upper-triangular integer matrices precisely when its diagonal entries are differences of two squares and its off-diagonal entry satisfies a divisib

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

The paper seeks to characterize completely the upper-triangular 2 by 2 integer matrices M that arise as A squared minus B squared for other upper-triangular integer matrices A and B. It links this to the classical fact that an integer is a difference of two squares exactly when it is not congruent to 2 modulo 4, but applies it to the diagonals p and q while imposing an extra divisibility requirement on r. A sympathetic reader would value this because it converts the search for matrix representations into a straightforward arithmetic check on the three entries. The authors further classify all such representable matrices using congruence conditions on p, q, and r. This work therefore makes the representability question decidable from the entries alone.

Core claim

We give a complete criterion in terms of representations of p and q as differences of two squares and an additional divisibility condition on r. We also give a complete classification of representable matrices in terms of congruence conditions on p, q, and r.

What carries the argument

The matrix equation A² - B² = M for upper-triangular A, B, and M, which imposes entrywise conditions reducing to representations as differences of squares on the diagonals plus a divisibility constraint on the off-diagonal term r.

If this is right

  • If p or q is congruent to 2 modulo 4, then the matrix cannot be represented as such a difference of squares.
  • When p and q are differences of squares and the divisibility holds for r, explicit A and B can be constructed.
  • The representable matrices are exactly those satisfying the listed arithmetic conditions on their entries.
  • Congruence conditions provide a complete list of all possible representable p, q, r triples.

Where Pith is reading between the lines

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

  • The same method of reducing matrix equations to entry conditions could apply to other forms like sums of squares or other matrix shapes.
  • This classification might help analyze the structure of the monoid of upper-triangular matrices under addition or multiplication.
  • Similar divisibility conditions may appear when considering representations over other rings or fields.

Load-bearing premise

The characterization depends on requiring A and B to be upper-triangular integer matrices whose squares differ exactly by M with no additional constraints or extensions allowed.

What would settle it

A concrete upper-triangular integer matrix M with p and q each equal to a difference of two squares, r satisfying the divisibility condition, but for which no upper-triangular integer matrices A and B exist such that A squared minus B squared equals M.

read the original abstract

We consider the problem of characterizing upper-triangular matrices $M=\begin{pmatrix}p&r\\0&q\end{pmatrix}\in M_2(\mathbb Z)$ which can be represented in the form $A^2-B^2$ with upper-triangular integer matrices $A$ and $B$ and give a complete criterion in terms of representations of $p$ and $q$ as differences of two squares and an additional divisibility condition on $r$. Also, we give a complete classification of representable matrices in terms of congruence conditions on $p$, $q$, and $r$.

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 / 2 minor

Summary. The paper characterizes upper-triangular 2×2 integer matrices M = [[p, r], [0, q]] that can be expressed as A² − B² for upper-triangular integer matrices A and B. It gives a complete criterion in terms of p and q each being a difference of two squares together with a divisibility condition on r, and an equivalent classification of representable matrices via congruence conditions on p, q, and r.

Significance. If the result holds, it supplies an explicit, complete characterization for this restricted class of matrices. Necessity of the conditions on p and q follows directly from the explicit expansion of A² − B² for upper-triangular A and B; the non-trivial content is the sufficiency of the stated divisibility condition on r. The work is of moderate interest in number theory as a model case for Diophantine representation problems in matrices.

minor comments (2)
  1. The abstract states the criterion exists but does not display the precise divisibility condition on r; including the exact statement would improve the preview for readers.
  2. A short section or paragraph with concrete numerical examples (including cases with p = 0 or q = 0) would help verify that the sufficiency proof covers the edge cases mentioned in the reader's note.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and recommending minor revision. The referee's summary accurately describes the main results: a complete characterization of upper-triangular 2×2 integer matrices that are differences of squares of similar matrices, via conditions on the diagonal entries being differences of squares together with a divisibility condition on the off-diagonal entry, and an equivalent formulation in terms of congruences.

Circularity Check

0 steps flagged

No circularity: direct characterization from matrix algebra

full rationale

The paper derives a complete criterion for when an upper-triangular integer matrix M = [[p, r], [0, q]] equals A² - B² for upper-triangular integer matrices A, B. Necessity of the difference-of-squares conditions on p and q follows immediately by expanding the (1,1) and (2,2) entries of A² - B²; the divisibility obstruction on r is likewise read off from the off-diagonal entry. The non-trivial content is the proof that the stated conditions are also sufficient, which is established by explicit construction of A and B rather than by any self-referential fit, renaming, or self-citation chain. No equation reduces to its own input by definition, no parameter is fitted and then relabeled a prediction, and the argument relies only on elementary arithmetic identities over the integers. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts about integer squares and divisibility; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Basic arithmetic properties of squares and divisibility in the integers
    The criterion is built on the known characterization of integers that are differences of squares together with divisibility rules.

pith-pipeline@v0.9.0 · 5399 in / 1234 out tokens · 41696 ms · 2026-05-12T00:51:21.622459+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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