Exploring Integer Solutions of Negative Pell’s Equation with Two-digit Keith Number

Abstract

This work explores non-trivial integer solutions to the negative Pell’s equation, specifically considering the two-digit Keith number. Furthermore, we identity interesting recurrence relations that emerge from these solutions.

Introduction

Summary:

The paper studies the negative Pell equation x2−Dy2=−1x^2 - D y^2 = -1x2−Dy2=−1, where DDD is a non-negative integer that is not a perfect square. A key necessary condition for this equation to have solutions is that DDD must not be divisible by 4 or by primes of the form 4k+34k+34k+3. For example, when D=3D = 3D=3, the equation has no solutions, but it may be solvable for other values of DDD.

The research focuses on exploring solutions to this negative Pell equation in relation to Keith numbers—special numbers that appear in sequences similar to the Fibonacci sequence, generated from their own digits (also called repfigit numbers), such as 14, 19, 28, 47, and so on.

Methodology:
Starting with an initial solution of the negative Pell equation, the study derives subsequent solutions by linking it to the classical Pell equation x2−Dy2=1x^2 - D y^2 = 1x2−Dy2=1. Using the Brahmagupta lemma, the paper generates a sequence of distinct non-zero integer solutions and formulates a recurrence relation to describe these solutions systematically.

Conclusion

This paper explores integer solutions to the negative Pell equation involving Keith number. The diverse nature of Diophantine equation, future research can extend this approach by investigating other types of Keith number within different Diophantine equations

References

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Copyright

Copyright © 2025 S Vidhya , N Priyabharathy. 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.