PreprintCC BY 4.0arXiv
D(N)-quadruples in upper-triangular 2×2 integer matrices
Abstract
We introduce analogues of Diophantine D(N)-m-tuples in the noncommutative ring M2(ℤ) of 2×2 integer matrices. Besides definitions based on the standard matrix product, we consider a symmetric version defined via the Jordan product A∘ B=1/2(AB+BA). Special attention is devoted to upper-triangular integer matrices UT2(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 UT2(ℤ).
Related
BibTeX
@misc{dujella2026dnquadruples,
title = {{$D(N)$-quadruples in upper-triangular $2\times2$ integer matrices}},
author = {Andrej Dujella and Zrinka Franušić},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29968},
eprint = {2608.29968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29968},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - D(N)-quadruples in upper-triangular 2×2 integer matrices
AU - Andrej Dujella
AU - Zrinka Franušić
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29968
UR - https://arxiv.org/abs/2608.29968
AB - We introduce analogues of Diophantine D(N)-m-tuples in the noncommutative ring M_2(ℤ) of 2×2 integer matrices. Besides definitions based on the standard matrix product, we consider a symmetric version defined via the Jordan product A∘ B=1/2(AB+BA). Special attention is devoted to upper-triangular integer matrices UT_2(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(ℤ).
ER -