Enduring Sequences of Hyperbolic Diophantine Quadruples involving Hilbert and Gnomonic Number

Abstract

This paper introduces Hyperbolic Diophantine quadruple, defined as set of four distinct positive integers satisfying the property D(n), where pairwise product plus n yields perfect square. A novel construction generates sequences of such quadruples by first forming a Hyperbolic pair using Hilbert number and Gnomonic number. Algebraic manipulations extend this pair to a triple, then to a quadruple via a conjecture ensuring the fourth element preserves D(n).

Introduction

The study extends the classical Diophantine D(n)D(n)D(n) problem, where a set of integers (a1,a2,…,aq)(a_1, a_2, …, a_q)(a1?,a2?,…,aq?) has the property D(n)D(n)D(n) if the product of any two distinct elements plus a fixed number nnn forms a perfect square. While early work by Diophantus found such sets for rational numbers, constructing integer tuples is more complex and has been widely researched.

This work introduces Hyperbolic Diophantine pairs using Hilbert numbers (4k+1)(4k+1)(4k+1) and Gnomonic numbers (2k−1)(2k-1)(2k−1), expressed with hyperbolic numbers of the form p1+p2jp_1 + p_2jp1?+p2?j where j2=1j^2 = 1j2=1. The study constructs pairs whose products plus a constant produce perfect squares, satisfying the D(4)D(4)D(4) or D(1)D(1)D(1) properties.

Using algebraic transformations and a conjecture stating that every Diophantine triple can be extended to a unique quadruple, the paper systematically extends:

• Pairs → Triples → Quadruples

Two main constructions are presented:

1. Hilbert-number based sequence producing Hyperbolic Diophantine quadruples with property D(4)D(4)D(4).

2. Gnomonic-number based sequence producing Hyperbolic Diophantine quadruples with property D(1)D(1)D(1).

In both cases, recursive methods allow the newly formed tuples to act as the basis for generating the next set, creating infinite sequences of Hyperbolic Diophantine quadruples while preserving the required D(n)D(n)D(n) condition.

Tables of numerical examples demonstrate these sequences for different values of parameters kkk and yyy.

Overall, the work introduces a new framework for generating infinite sequences of Hyperbolic Diophantine quadruples by combining hyperbolic numbers with Hilbert and Gnomonic number structures.

Conclusion

This paper establishes a systematic framework for generating infinite sequences of Hyperbolic Diophantine quadruples through algebraic manipulations, presenting concrete results. Future research directions include extension to higher m-tuples, and exploration in cryptographic applications.

References

[1] Apostol, Tom M. Introduction to Analytic Number Theory. Springer Science & Business Media, 2013. [2] Dickson, Leonard Eugene. History of the Theory of Numbers. Vol. 1, University of Pennsylvania Press, 1999. [3] Dujella, Andrej. Diophantine m-Tuples and Elliptic Curves. Springer, Cham, 2024. [4] Dujella, Andrej, and A. M. S. Ramasamy. “Fibonacci Numbers and Sets with the Property D(4).” Bulletin of the Belgian Mathematical Society – Simon Stevin, vol. 12, no. 3, 2005, pp. 401–412. [5] Fujita, Yasutsugu, and Takafumi Miyazaki. “The Regularity of Diophantine Quadruples.” Transactions of the American Mathematical Society, vol. 370, no. 6, 2018, pp. 3803–3831. [6] Geetha, V., B. Priyanka, and N. Harini. “A Study on Some Sequences of Gaussian Diophantine Triples.” Arya Bhatta Journal of Mathematics and Informatics, vol. 16, no. 1, 2024, pp. 71–76. [7] Janaki, G., and C. Saranya. “Gaussian Diophantine Quadruples Involving Gnomonic Numbers with Property D(4).” Journal Title Unknown, vol. 5, 2019, pp. 341–342. [8] Janaki, G., and R. Sarulatha. “On Sequences of Geophine Triples Involving Padovan and Bernstein Polynomial with Propitious Property.” Indian Journal of Science and Technology, vol. 17, no. 16, 2024, pp. 1690–1694. [9] Koblitz, Neal. A Course in Number Theory and Cryptography. Vol. 114, Springer Science & Business Media, 1994. [10] Niven, Ivan, Herbert S. Zuckerman, and Hugh L. Montgomery. An Introduction to the Theory of Numbers. John Wiley & Sons, 1991. [11] Pandichelvi, V., and B. Umamaheswari. “Establishment of Order of 4-Tuples Concerning Familiar Polynomials with Captivating Condition.” Indian Journal of Science and Technology, vol. 16, no. 36, 2023, pp. 2982–2987. [12] Saranya, P., and G. Janaki. “Gaussian Quadruples Involving Fermat Number, Stella Octangula Number and Pronic Number.” Infokara Research Journal, vol. 9, no. 1, 2020, pp. 955–958. [13] Saranya, P., and G. Janaki. “On Some Non-Extendable Gaussian Triples Involving Mersenne and Gnomonic Number.” Arya Bhatta Journal of Mathematics and Informatics, vol. 12, no. 1, 2020, pp. 81–84. [14] Trebješanin, Marija Bliznac, and Sanda Buja?i? Babi?. “Polynomial Quadruples over Gaussian Integers.” GlasnikMatemati?ki, vol. 59, no. 1, 2024, pp. 1–31.

Copyright

Copyright © 2026 R. Sarulatha , G. Janaki. 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.