On a Kind of Nonlinear Diophantine Equation

TL;DR

This paper generalizes linear Diophantine equations to a nonlinear form involving determinants, providing necessary and sufficient conditions for solvability and illustrating how linear equations are special cases.

Contribution

It introduces a nonlinear Diophantine equation involving determinants and establishes criteria for its solvability, extending classical linear results.

Findings

Derived necessary and sufficient conditions for solutions

Showed how linear equations are special cases

Provided a framework for analyzing nonlinear Diophantine equations

Abstract

In this paper we show a way to generalize the linear Diophantine equation a1x1+a2x2+...+anxn=d . We deal with the nonlinear Diophantine equation det|A X|=+-d , which generalizes the linear one, and we give a necessary and sufficient condition for its solvabiliy. We show how the linear equation can be considered as a particular case of the nonlinear equation.