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.
Citation: Hongjian Li, Pingzhi Yuan. Fermat's and Catalan's equations over $ M_2(\mathbb{Z}) $[J]. AIMS Mathematics, 2023, 8(1): 977-996. doi: 10.3934/math.2023047
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.
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Hongjian Li, Pingzhi Yuan. Fermat's and Catalan's equations over $ M_2(\mathbb{Z}) $[J]. AIMS Mathematics, 2023, 8(1): 977-996. doi: 10.3934/math.2023047
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