## Saturday, November 15, 2014

### Preparing for Fibonacci Day - Pisano Periods

Field Trip!  Go watch the Numberphile Video: http://www.numberphile.com/videos/pisano_period.html
The Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
Many mathematicians have studied this integer sequence for hundreds of years and found several interesting patterns.  One of them is a pattern called Pisano periods.
It says that the mth Fibonacci number (or F(m)) evenly divides the nth Fibonacci number (or F(n)) if m evenly divides n.  So the 3rd Fibonacci number (or F(3) which equals 2) divides every 3rd Fibonacci number ((F(6), F(9), F(12), etc.)
 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 2 is the 3rd Fibonacci number.  Every 3rd number after 2 is divisible by 2 (it’s and even number). 3 is the 4th Fibonacci number.  Every 4th number after 3 is divisible by 3. 5 is the 5th Fibonacci number.  Every 5th number after 5 is divisible by 5. 8 is the 6th Fibonacci number.  Every 6th number after 8 is divisible by 8. 13 is the 7th Fibonacci number.  Every 7th number after 13 is divisible by 13. 21 is the 8th Fibonacci number.  Every 8th number after 21 is divisible by 21. 34 is the 9th Fibonacci number.  Every 9th number after 34 is divisible by 34. 55 is the 10th Fibonacci number.  Every 10th number after 55 is divisible by 55. 89 is the 11th Fibonacci number.  Every 11th number after 89 is divisible by 89. 144 is the 12th Fibonacci number.  Every 12th number after 144 is divisible by 144. Etc.

David