From Diophantian Equations to Matrix Equations (Iv) - Diophantian Equations of Higher Degree

In the context of training and developing the skills of teachers, students and children to solve exercises and problems in Mathematics, in this paper we propose to continue the steps started in the first three papers with the same generic title and marked symbolically with (I), (II), respectively with (III) namely, the presentation of some ways of achieving a systemic vision on a certain notional mathematical content, a vision that motivates and mobilizes the activity of those who teach in the classroom, thus facilitating both the teaching and the assimilation of the concepts, concepts, and scientific theories addressed of the educational disciplines that present phenomena and processes from nature. Thus, we will continue in the same systemic approach, solving some Diophantine equations of higher degree, more precisely some generalizations of the Pythagorean equation and some quadratic Diophantine equations, in the set of natural numbers, then of the whole numbers, in order to "submerge" a such an equation in a ring of matrices and try to find as many matrix solutions as possible. The methods used, in this work, in solving the Diophantine equations, are diversified and different, for the most part, from those used in previous works of this type. It will result, then, that the methods of solving matrix equations related to these equations will follow the same direction. To facilitate the reader's understanding of the solutions of these matrix equations, in some cases we have also presented examples.