Fermat's type equations in the set of \(2\times 2\) integral matrices (Q1293147)

Let \((*)\) be \(X^n+ Y^n= Z^n\) and \((**)\) be \(X^n+ Y^n+ Z^n= W^n\) \((n\geq 1)\). Let \(G(k,m)= \{M\); \(\text{det}(M)= m\}\), where \(M= \left( \begin{smallmatrix} r&b\\ ks&r \end{smallmatrix} \right)\), \(r,s\in \mathbb{Z}\); \(k\) a fixed positive integer not a perfect square. The following theorems are proved: 1. \((*)\) has no solutions in \(X,Y,Z\in G(k,\pm 1)\) for any positive integer \(n\). 2. Let \(a\) be a fixed integer. If \(X,Y,Z\in G(k,a)\), then \((*)\) with positive integer exponent \(n\geq 3\) does not hold except when \(X=0\) or \(Y=0\) or \(Z=0\). 3. If \(X,Y,Z,W\in G(k,a)\) and \(k>1\) is a fixed square-free integer, then the equation \((**)\) does not hold except when \(X+Y=0\) or \(Y+Z=0\) or \(Z+X=0\) and \((n,2)=1\).