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This is a Quantitative Study study.

September 17, 2025Open Access

THE ROLE OF CONTINUED FRACTIONS IN SOLVING PELL-TYPE EQUATIONS VIA (γ,δ)-JACOBSTHAL AND(γ,δ)-JACOBSTHAL-LUCAS SEQUENCES

BUB. UmamaheswariMeenakshi Academy of Higher Education and Research VPV. Pandichelvi

Key Points

Solutions are articulated through (γ,δ)-jacobsthal and (γ,δ)-jacobsthal lucas sequences, enhancing existing methodologies.
Using principles from number theory, the research covers two categories of diophantine equations with specialized primes.
The study utilizes Python codes to confirm the discovered solutions for the equations under exploration.
Findings may influence further developments in resolving other complex diophantine equations and prime classifications.

Abstract

In this manuscript, the techniques for finding the solutions in terms of (γ, δ) - Jacobsthal and (γ, δ) - Jacobsthal Lucas sequences for two distinct categories of the Diophantine equations uₙ²- (p²±q) vₙ²=q²n and uₙ²- (4p² q²-8) vₙ²= (-2δ) ⁿ, n≥1, δ∈Z-0 where p and q are specialised prime numbers, including Gaussian prime, Good prime, Lucky prime, Chen prime and Circular prime are offered. Moreover, the entire solutions discovered are confirmed with the assistance of three distinct Python codes. KEYWORDS: (γ, δ) - Jacobsthal, (γ, δ) - Jacobsthal Lucas, Gaussian prime, Good prime, Lucky prime, Chen prime, Circular prime.

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