Investigating Fibonacci Sequence Properties for Theoretical Computer Science

The Fibonacci sequence has played dominant roles in many branches of science and engineering. This sequence not only has an amazing number of applications in nature, but also has a tremendous number of interesting properties. The main object of this paper is to propose a systematic investigation of the familiar Fibonacci sequence in number theory and present a heuristic derivation of their properties. In this paper, we not only analyze the basic concepts and definitions of recurrence functions, generating functions, continued fractions, the Golden ratio, and primality test, but also give detailed analysis and further discussions for them with a different point of view when they incorporated with the Fibonacci sequence. Based on these concepts and definitions, we engage to contribute exact theorem descriptions and proofs. Moreover, extended properties including recurrence relations, summation formulas, and primality proving issues related to Fibonacci sequence are systematically studied and further explored.

KEYWORDS:Algorithmic number theory, elementary cryptography, information science, primality proving, recurrence relation.