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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.23404 |
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| _version_ | 1866909027794944000 |
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| author | Dujella, Andrej Franušić, Zrinka |
| author_facet | Dujella, Andrej Franušić, Zrinka |
| contents | 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 $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23404 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Differences of squares of upper-triangular $2\times 2$ integer matrices Dujella, Andrej Franušić, Zrinka Number Theory 11C20, 15A24, 11A07 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 $r$. |
| title | Differences of squares of upper-triangular $2\times 2$ integer matrices |
| topic | Number Theory 11C20, 15A24, 11A07 |
| url | https://arxiv.org/abs/2604.23404 |