Fermat's and Catalan's equations over $ M_2(\mathbb{Z})

Let $ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in M_2\left(\mathbb{Z}\right) $ be a given matrix such that $ bc\neq0 $ and let $ C(A) = \{B\in M_2(\mathbb{Z}): AB = BA\} $. In this paper, we give a necessary and sufficient condition for the solvability of the matrix equation $ uX^i+v...

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Veröffentlicht in:AIMS mathematics Jg. 8; H. 1; S. 977 - 996
Hauptverfasser: Li, Hongjian, Yuan, Pingzhi
Format: Journal Article
Sprache:Englisch
Veröffentlicht: AIMS Press 01.01.2023
Schlagworte:
catalan's matrix equation
fermat's matrix equation
matrix equation
quadratic fields
ISSN:2473-6988, 2473-6988
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Zusammenfassung:Let $ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in M_2\left(\mathbb{Z}\right) $ be a given matrix such that $ bc\neq0 $ and let $ C(A) = \{B\in M_2(\mathbb{Z}): AB = BA\} $. In this paper, we give a necessary and sufficient condition for the solvability of the matrix equation $ uX^i+vY^j = wZ^k, \, i, \, j, \, k\in\mathbb{N}, \, X, \, Y, \, Z\in C(A) $, where $ u, \, v, \, w $ are given nonzero integers such that $ \gcd\left(u, \, v, \, w\right) = 1 $. From this, we get a necessary and sufficient condition for the solvability of the Fermat's matrix equation in $ C(A) $. Moreover, we show that the solvability of the Catalan's matrix equation in $ M_2\left(\mathbb{Z}\right) $ can be reduced to the solvability of the Catalan's matrix equation in $ C(A) $, and finally to the solvability of the Catalan's equation in quadratic fields.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2023047