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arXiv · 2608.29968

$D(N)$-quadruples in upper-triangular $2\times2$ integer matrices

Andrej Dujella·Zrinka Franušić
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Abstract

We introduce analogues of Diophantine $D(N)$-$m$-tuples in the noncommutative ring $M_2(\mathbb Z)$ of $2\times2$ integer matrices. Besides definitions based on the standard matrix product, we consider a symmetric version defined via the Jordan product $$A\circ B=\frac12(AB+BA).$$ Special attention is devoted to upper-triangular integer matrices $UT_2(\mathbb Z)$, where squares admit a particularly simple description. Motivated by the classical connection between representations of $n$ as a difference of two squares and the existence of $D(n)$-quadruples in commutative rings, we investigate the existence of Jordan $D(N)$-quadruples in $UT_2(\mathbb Z)$.

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Andrej Dujella, Zrinka Franušić. 2026-08-30. $D(N)$-quadruples in upper-triangular $2\times2$ integer matrices. https://arxiv.org/abs/2608.29968

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