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- W4313443086 abstract "Let $mathbb{N}$ be the set of all positive integers and let $a,, b,, c$ be nonzero integers such that $gcdleft(a,, b,, cright)=1$. In this paper, we prove the following three results: (1) the solvability of the matrix equation $aX^m+bY^n=cI,,X,,Yin M_2(mathbb{Z}),, m,, ninmathbb{N}$ can be reduced to the solvability of the corresponding Diophantine equation if $XYneq YX$ and the solvability of the equation $ax^m+by^n=c,, m,, ninmathbb{N}$ in quadratic fields if $XY=YX$; (2) we determine all non-commutative solutions of the matrix equation $X^n+Y^n=c^nI,,X,,Yin M_2(mathbb{Z}),,ninmathbb{N},,ngeq3$, and the solvability of this matrix equation can be reduced to the solvability of the equation $x^n+y^n=c^n,, ninmathbb{N},,ngeq3$ in quadratic fields if $XY=YX$; (3) we determine all solutions of the matrix equation $aX^2+bY^2=cI,,X,,Yin M_2(mathbb{Z})$." @default.
- W4313443086 created "2023-01-06" @default.
- W4313443086 creator A5046863635 @default.
- W4313443086 creator A5065755582 @default.
- W4313443086 date "2022-12-28" @default.
- W4313443086 modified "2023-10-18" @default.
- W4313443086 title "The matrix equation $aX^m+bY^n=cI$ over $M_2(mathbb{Z})$" @default.
- W4313443086 doi "https://doi.org/10.48550/arxiv.2212.14139" @default.
- W4313443086 hasPublicationYear "2022" @default.
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