Fermat's problem and Goldbach's problem over \(M_ n \mathbb{Z}\) (Q1911424)

The author is looking for solutions of the Fermat equation \(x^m+ y^m= z^m\) with unknowns \(x\), \(y\), \(z\) being integral two by two matrices of determinant 1. Solving an open problem of the reviewer, the author proves that such a solution exists if and only if \(m\) is not divisible by 3 or 4. The same result is contained in a paper by A. Khazanov and a paper by Zhang. The author also considers Goldbach's problem over integral matrices [see also Math. Mag. 692 (1996)].