Solutions of Pell’s Equation Involving Wilson’s Primes

Abstract

The main goal of this study is to find non-trivial integer solutions to Pell’s equation involving Wilson’s primes. We obtain a series of unique integer solution and construct recurrence linkage between them using Brahmagupta’s lemma. The derived solutions and their relationships are shown numerically, illustrating the function of Wilson’s primes in producing these solutions.

Introduction

A key topic in number theory is the study of Diophantine equations, which seek integer solutions to polynomial equations. One of the most important examples is Pell’s equation, where DDD is a positive square-free integer. This equation is significant because it has strong connections with quadratic fields, continued fractions, and algebraic number theory, and it generally admits infinitely many integer solutions once a fundamental solution is known.

Historically, mathematicians have developed several methods to solve Pell’s equation, including continued fraction techniques and algebraic approaches. A classical and powerful method is Brahmagupta’s lemma, which provides a composition rule that allows new solutions to be generated from known ones. This lemma reveals the multiplicative structure of Pell’s equation and produces recurrence relations that describe how solutions grow.

Recent research explores the use of special prime numbers, particularly Wilson’s primes, to construct structured solutions for Diophantine equations. Wilson’s primes are rare primes that satisfy a special congruence derived from Wilson’s theorem. Their unique arithmetic properties make them useful for generating specific patterns of integer solutions.

This study investigates Pell’s equation by combining Wilson’s primes with Brahmagupta’s lemma. By using the special properties of Wilson’s primes, the authors derive recurrence relations and demonstrate the existence of distinct non-zero integer solutions to the equation. The theoretical results are supported with numerical examples, contributing to a deeper understanding of structured solution generation in classical Diophantine equations.

Conclusion

In this paper, we have presented a structured method for generating non-zero and distinct integer solutions to Pell’s equation by incorporating the arithmetic properties of Wilson’s primes. By applying Brahmagupta’s lemma as a fundamental tool, we established recurrence relations that systematically produce infinite sequences of solutions from an initial solution pair. The factorial congruence condition characterizing Wilson’s primes plays a crucial role in constructing these solution families and in strengthening the underlying algebraic structure of the equation. The results demonstrate that the classical theory of Pell’s equation can be significantly enriched through the integration of special prime classes such as Wilson primes. This interplay between prime number theory and quadratic Diophantine equations not only deepens the understanding of solution structures but also opens new directions for further research in the study of Pell-type equations and related number-theoretic problems.

References

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Copyright

Copyright © 2026 C. Saranya, M. Shalini. 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.