Current Issue

Sardar Patel International Journal of Interdisciplinary Research & Innovation
ISSN: 3107-832X (Online)

Properties and Applications of Generalized Pell Equations

ABSTRACT

Generalized Pell equations represent a class of Diophantine equations in which a quadratic expression is related to a constant or to a parameter. The linear equation (x2 - Dy2 = 1) is the chief representative of this class and it is well known for its interconnection between each area of elementary number theory, continued fraction theory, theory of recurrence relations, and algebraic number theory. Generalized forms of the Pell equation are considered in this article. Some of which include the general created form (x2 - Dy2 = N) and other related forms with some parameterization variation. The objective is the study of aspects of the existence of solutions, the generation of solutions using the theory of recurrences, periodicity in the representation of solutions using the generalized continued fraction, and various computational techniques. The connection to Pell equations and the systems of the units of quadratic fields is illustrated. A number of examples are presented to show how solutions to Pell equations may be generated. Some applications in the development of approximation ‘theory’, the creation of cryptographic protocols, the enumeration of combinatorial consequences of recurrences, and algorithmic number theory are considered. The paper concludes with a discussion of the means to implement a set of related computational methods to build solutions to higher ‘degree’ problems and systems of coupled Diophantine equations.

FULL TEXT