Divisibility and Periodicity in the Fibonacci Sequence

摘要

The Fibonacci sequence F_n is defined recursively as follows: F_1 = F_2 = 1, F_n =F_(n-2) + F _ (n-1), for n ≥ 3. When it is required, we define F_0 = 0. It is well known that every third Fibonacci number is even and every fifth Fibonacci number is divisible by 5. In contrast, some easily derived results may be less well known including that no odd Fibonacci number is divisible by certain primes including 17 and 61 and no Fibonacci number is congruent to either 4 modulo 8 or 6 modulo 8. Another classical result asserts that one of the first n~2 terms is divisible by n . Our goal is to illustrate how modular arithmetic can be used in discovering signs of divisibility and periodicity ideas in the Fibonacci sequence. As a postscript, we will briefly view the Lucas sequence as a foil to the Fibonacci sequence.