论文标题
矩阵方程$ ax^m+by^n = ci $ of $ m_2（\ mathbb {z}）$
The matrix equation $aX^m+bY^n=cI$ over $M_2(\mathbb{Z})$
论文作者
论文摘要
令$ \ mathbb {n} $为所有正整数的集合，让$ a，\，b，\，c $为非零整数，以至于$ \ gcd \ left（a，\，b，b，\，c \ right）= 1 $。在本文中，我们证明了以下三个结果：（1）矩阵方程$ ax^m+by^n = ci，\，\，\，x，\，y \ in m_2（\ mathbb {z}），\，\，m，m，\ \，n \，in \ mathbb {n} $可以减少solity n e soliation n for solage in for soly sighitiation die di. yx $和等式$ ax^m+by^n = c，\，m，\，n \ in \ mathbb {n} $在二次字段中，如果$ xy = yx $; （2）我们确定矩阵方程的所有非交换解决方案$ x^n+y^n = c^n = c^ni，\，\，x，\，\，y \ in m_2（\ mathbb {z}），\，n \ in \ mathbb i {n}，n}，\ geq3 $，以及该方程式的求解方式， $ x^n+y^n = c^n，\，n \ in \ mathbb {n}，\，\，n \ geq3 $在二次字段中，如果$ xy = yx $; （3）我们确定矩阵方程的所有解决方案$ ax^2+by^2 = ci，\，\，x，\，y \ in m_2（\ mathbb {z}）$。
Let $\mathbb{N}$ be the set of all positive integers and let $a,\, b,\, c$ be nonzero integers such that $\gcd\left(a,\, b,\, c\right)=1$. In this paper, we prove the following three results: (1) the solvability of the matrix equation $aX^m+bY^n=cI,\,X,\,Y\in M_2(\mathbb{Z}),\, m,\, n\in\mathbb{N}$ can be reduced to the solvability of the corresponding Diophantine equation if $XY\neq YX$ and the solvability of the equation $ax^m+by^n=c,\, m,\, n\in\mathbb{N}$ in quadratic fields if $XY=YX$; (2) we determine all non-commutative solutions of the matrix equation $X^n+Y^n=c^nI,\,X,\,Y\in M_2(\mathbb{Z}),\,n\in\mathbb{N},\,n\geq3$, and the solvability of this matrix equation can be reduced to the solvability of the equation $x^n+y^n=c^n,\, n\in\mathbb{N},\,n\geq3$ in quadratic fields if $XY=YX$; (3) we determine all solutions of the matrix equation $aX^2+bY^2=cI,\,X,\,Y\in M_2(\mathbb{Z})$.
