On the Solutions of det(A/X)=+-d

TL;DR

This paper investigates a non-linear Diophantine equation derived from the determinant of an integer matrix, providing methods to find solutions, classify them into equivalence classes, and compute their total number.

Contribution

It introduces a novel approach to solving a specific determinant-based Diophantine equation, including solution formulas, classification, and an algorithm.

Findings

Solutions can be explicitly characterized and partitioned into finite equivalence classes.

Formulas for all solutions and their count are derived.

An algorithm for computing solutions is proposed and demonstrated.

Abstract

In this paper we deal with a non-linear Diophantine equation which arises from the determinant computation of an integer matrix. We show how to find a solution, when it exists. We define an equivalence relation and show how the set of all the solutions can be partitioned in a finite set of equivalence classes and find a set of solutions, one for each of these classes. We find a formula to express all the solutions and a formula to compute the cardinality of the set of fundamental solutions. An algorithm to compute the solutions is proposed and clarified with some examples.