From Diophantian Equations to Matrix Equations (III) - Other Diophantian Quadratic Equations and Diophantian Equations of Higher Degree

Abstract

In this paper, we propose to continue the steps started in the first two papers with the same generic title and symbolically denoted by (I) and (II), namely, the presentation of ways of achieving a systemic vision on a certain mathematical notional content, a vision that to motivate and mobilize the activity of those who teach in the classroom, thus facilitating both the teaching and the assimilation of notions, concepts, scientific theories approached by 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. In this way we will solve 12 large classes of Diophantine quadratic or higher order equations. For attentive readers interested in these matters, at the end of the paper we will propose 6 open problems. The solution of each of these open problems represents, in fact, a vast research activity and that can open the way to solving such more complicated Diophantine and / or matrix equations.