A Mathematical Analysis of the Lucas Sequence and its Relationship to Prime Number Properties and the Silver Ratio in Light of Number Theory

Abstract

This study presents a mathematical analysis of the Lucas sequence and its relationship to prime number testing algorithms and the silver ratio within the framework of number theory. It aims to highlight the unique numerical properties of this sequence and explore the patterns inherent in its terms and their numerical convergence. The study employs a descriptive-analytical approach, drawing on theoretical sources and specialized textbooks. Computer programs such as Mathematica and MATLAB were used to analyze the behavior of the sequence's terms and observe their convergence towards the silver ratio. The procedures included numerical and graphical analysis of a sample of terms and linking the results to the principles of number theory to test for the emergence of prime numbers and extract patterns.The results showed that the Lucas sequence possesses precise numerical properties, and that some of its terms include prime numbers that can be utilized in developing prime-testing algorithms. The study also demonstrated that its terms gradually converge towards the silver ratio and exhibit recurring numerical patterns of mathematical and applied significance. The study recommended expanding future research on the Lucas sequence and linking it to deeper applications in cryptography and computer science, along with conducting comparative studies with similar sequences to enhance theoretical and applied understanding.