The matrix equation aX^m+bYⁿ=cI over M₂(ℤ)

2022/12/29 by Li, Hongjian, Yuan, Pingzhi
#11D09 #11D41 #15A20 #15A24 #15B36 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2212.14139

Abstract

Let ℕ be the set of all positive integers and let a, b, c be nonzero integers such that gcd(a, b, c)=1. In this paper, we prove the following three results: (1) the solvability of the matrix equation aX^m+bYⁿ=cI, X, Y∈ M₂(ℤ), m, n∈ℕ can be reduced to the solvability of the corresponding Diophantine equation if XY≠ YX and the solvability of the equation ax^m+byⁿ=c, m, n∈ℕ in quadratic fields if XY=YX; (2) we determine all non-commutative solutions of the matrix equation Xⁿ+Yⁿ=cⁿI, X, Y∈ M₂(ℤ), n∈ℕ, n≥3, and the solvability of this matrix equation can be reduced to the solvability of the equation xⁿ+yⁿ=cⁿ, n∈ℕ, n≥3 in quadratic fields if XY=YX; (3) we determine all solutions of the matrix equation aX²+bY²=cI, X, Y∈ M₂(ℤ).

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