Fibonacci Numbers

Fibonacci, also known as Leonardo of Pisa, was one of the greatest European mathematicians of the middle ages. His book on arithmetic played a major role in the spreading of Hindu-Arabic numerals into Europe. In 1202, he posed the following problem:

A male and a female rabbit are born at the beginning of the year. We assume the following conditions:

Rabbit pairs are not fertile during their first month of life, but thereafter give birth to one new male/female pair at the end of every month.

No deaths occur during the year.

How many rabbits will there be at the end of the year?

If we let *F(n)* denote
the number of rabbit pairs at the beginning of month *n*,
then we have

*F(1)=1, F(2)=1, F(3)=2, F(4)=3, F(5)=5,...*

and,in general,

*F(n+2)=F(n)+F(n+1).*

The sequence *{1,1,2,3,5,8,...} *generated
according to this scheme was named in honor of Fibonacci. The
Fibonacci sequence was one of the earilest examples of a
recursively defined sequence. There are many other examples in
population dynamics where the Fibonacci sequence arises.

The first 55 Fibonacci numbers are given in the following table:

n, F(n) |
n, F(n) |
n, F(n) |
n, F(n) |
n, F(n) |

1, 1 | 12, 144 | 23, 28657 | 34, 5702887 | 45, 1134903170 |

2, 1 | 13, 233 | 24, 46368 | 35, 9227465 | 46, 1836311903 |

3, 2 | 14, 377 | 25, 75025 | 36, 14930352 | 47, 2971215073 |

4, 3 | 15, 610 | 26, 121393 | 37, 24157817 | 48, 4807526976 |

5, 5 | 16, 987 | 27, 196418 | 38, 39088169 | 49, 7778742049 |

6, 8 | 17, 1597 | 28, 317811 | 39, 63245986 | 50, 12586260925 |

7, 13 | 18, 2584 | 29, 514229 | 40, 102334155 | 51, 20365011074 |

8, 21 | 19, 4181 | 30, 832040 | 41, 165580141 | 52, 32951280099 |

9, 34 | 20, 6765 | 31, 1346269 | 42, 267914296 | 53, 53316291173 |

10, 55 | 21, 10946 | 32, 2178309 | 43, 433494437 | 54, 86267571272 |

11, 89 | 22, 17711 | 33, 3524578 | 44, 701408733 | 55, 139583862445 |

The first 500 Fibonacci numbers can be found here.

- On many flowers, the number of petals is a Fibonacci number.
- Fibonacci numbers can also be seen in the arrangement of seeds on flowerheads.
- Many plants show the Fibonacci numbers in the arrangements of the leaves around the stems.

- Consecutive Fibonacci numbers give the worst case behavior when they are used as inputs in the Euclidean algorithm. (Recall that the Euclidean algorithm is used to find the GCD of two positive integers.)
- As
*n*approaches infinity, the ratio*F(n+1)/F(n)*approaches the golden ratio (*f=1.6180339887498948482*...). The Greeks felt that rectangles whose sides are in the golden ratio are aesthetically most pleasing. - The Fibonacci number
*F(n+1)*gives the number of ways for*2 x 1*dominoes to cover a*2 x n*checkerboard. - The sum of the first
*n*Fibonacci numbers is*F(n+2)-1*. - The "shallow" diagonals of Pascal's triangle sum to Fibonacci numbers.
- (Except for the case
*n=4*) If*F(n)*is prime, then*n*is prime. Equivalently, if*n*is not prime, then*F(n)*is not prime. *gcd( F(n), F(m) ) = F( gcd(n,m) )*

A great deal of information is available on the Fibonacci numbers. Here are some excellent sites to visit:

The Fibonacci Numbers and the Golden section

Fibonacci - A biography.

Fibonacci - Another biography.

*Steve
Kifowit*

Prairie State College

Chicago Heights, IL