Development and prospects of Stewart’s theorem research

Abstract

This paper is devoted to the study of Stewart's theorem, its consequences, development and prospects of research of the theorem under consideration. The paper focuses on the Diophantine equations and their relation to the Stewart’s theorem. The problem of determination of integer solutions of the Diophantine equations was considered, and some modern researches of the Stewart’s theorem are presented from the point of view of finding integer solutions of it, which are related to the first and second order Diophantine equations. The wellknown integer solutions of the Stewart’s theorem, and the definitions, which have been formulated in the form of a table, are presented in this paper. Some practical applications of the Stewart’s theorem focused on computing the length of a segment, that connects the vertex of a triangle with its inner point, are relevant in the area of logistics, management and designing.