Pellian Equation Involving Factorian Number

Abstract

Thepositive Pellianbinary quadratic equation is studied for its distinct integer solutions, along with an analysis of several interesting relationships among them. Furthermore, by utilizing the solutions varioushyperbolas, parabolas, and special Pythagorean triangles are generated.

Introduction

Overview

The Pell equation is a classic Diophantine equation of the form:

x2−Dy2=1x^2 - Dy^2 = 1x2−Dy2=1

where D is a positive, square-free integer. This equation has fascinated mathematicians since antiquity. J.L. Lagrange proved that such equations always have integer solutions for any non-square positive integer D.

Applications

• Number theory

• Cryptography

• Quadratic forms

• Algebraic number theory

Related Concept: Factorion Numbers

A factorion is a number where the sum of the factorials of its digits equals the number itself, e.g., 1, 2, 145, 40585.

Method of Analysis

• The study explores the positive Pell equation and proves the existence of infinitely many solutions.

• Using Brahmagupta's lemma, additional solutions are derived from the smallest one.

• Solutions follow a recurrence relation for both x and y values.

• Example values for D = 145:

Key Mathematical Properties

Several expressions involving x and y from the Pell solution are identified as:

• Perfect squares

• Nasty numbers (presumably, numbers with special, complex properties)

• Cubical integers (perfect cubes)

Remarkable Observations

• Linear combinations of Pell solutions can generate solutions for other hyperbolas.

• Similar relationships are also observed in the context of parabolas.

Conclusion

This paper examines integer solutions to the positive Pell equation involving Factorian numbers. From the diverse nature of Diophantine equations, future research can build on this approach by investigating various types of Factorian numbers within different mathematical contexts.

References

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Copyright

Copyright © 2025 P. Saranya, B. Umadevi . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.