Mathematical Discovery: The Fibonacci Sequence

Mathematicians explore quantitative relationships that occur naturally in the real world. Here, we'll have a look at the Fibonacci sequence, a numerical sequence that shows up in a surprisingly large variety of settings.

Mathematical Discovery: The Fibonacci Sequence
A man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?   (Liber Abaci, c. 1202)
Leonardo Pisano Fibonacci
1170 -- 1250
Beginning of month 1
Beginning of month 2
Beginning of month 3
Beginning of month 4
Beginning of month 5
Beginning of month 6
1, 1, 2, 3, 5, 8, ___, ___, ...
The Fibonacci sequence
"Recurrence relation"
PRIME number = Positive whole number (> 1) whose only divisors are 1 and itself. 
Which Fibonacci numbers are prime numbers?
How many Fibonacci primes are there?
Status: OPEN 
What is the largest known Fibonacci prime?
Discovered June 2009; 376817 digits
Which Fibonacci numbers are even numbers?
Multiples of 3?
Multiples of 5?
Multiples of 8?
What is the pattern?
Take the Fibonacci numbers and divide by 2, keep the remainder.
What pattern do you see? 

What if you divide by 3?  By 4? By 5? etc.? 
Pisano cycles
Formula for the Pisano cycle for a given divisor: UNKNOWN 
Look at divisors giving Pisano cycles whose lengths are powers of 2 (2, 4, 8, 16, 32, etc.) 
What happens when we divide consecutive Fibonacci numbers? 
The Golden Ratio
Mathematics is about discovering relationships and patterns that exist in the world around us.