The Period of the Fibonacci Sequence Modulo j-斐波那契数列取模循环节.pdfVIP

The Period of the Fibonacci Sequence Modulo j Charles W. Campbell II Math 399 Spring 2007 Advisor: Dr. Nick Rogers Introduction The main objectives of this project were to examine the Fibonacci sequence mod- ulo j , a positive integer, with the intentions of generalizing the results to general se- quences defined by linear recursions and to utilize information being acquired during the semester. The main part of the research involved investigating the periodicity of the new sequence obtained after modding out by j . We begin by establishing a moti- vation for investigating the sequence modulo p a prime and pk a prime power by first investigating the Fibonacci sequence modulo j , a positive integer. Next we investigate the Fibonacci sequence modulo p a prime and then generalize to prime powers. True to the objective the results we obtained were through the use of (but not limited to) important ideas from both Number Theory and Abstract Algebra including Fermat’s Little Theorem , Euler’s generalization of Fermat’s Little Theorem, Quadratic Reci- procity, and Field Theory. Lastly we apply the results established for the Fibonacci sequence to easily prove results for the Lucas sequence modulo p a prime. We begin by defining the sequence itself. Definition The Fibonacci sequence is a linear recursion defined by Fn+1 = Fn−1 + Fn for n ≥ 1, (1) where Fn is the nth Fibonacci number with F0 = 0 and F1 = F2 = 1. In the study of the Fibonacci sequence, it will be nice to be able to calculate the Fibonacci numbers themselves. There is a closed form equation for doing just that, but befor