# Fibonacci number

In the text*Liber Abaci*written by Fibonacci in 1202, the following question was posed.

A man puts one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year, if the nature of these rabbits is such that every month each pair bears a new pair which from the second month becomes productive?The number of pairs of rabbits in the

*n*th month begins 1, 1, 2, 3, 5, 8, 13, 21, ... (often denoted

*u*

_{1},

*u*

_{2},

*u*

_{3}...) where each term is the sum of the two terms preceding it. Mathematicians define this sequence recursively as follows:

This sequence, now called the Fibonacci sequence, has an amazing number of applications in nature and art; it also has a tremendous number of interesting properties--which is reason enough for the journal "The Fibonacci Quarterly" to exist! Here we list just a few of these many properties. Letu_{1}=u_{2}= 1 andu_{n+1}=u_{n}+u_{n-1}(n> 2)

*m*and

*n*be positive integers,

*u*_{n}divides*u*_{mn},- gcd(
*u*_{n},*u*_{m}) =*u*_{gcd(m,n)}, *u*_{n}^{2}-*u*_{n+1}*u*_{n-1}= (-1)^{n-1},*u*_{1}+*u*_{3}+*u*_{5}+ ... +*u*_{2n-1}=*u*_{2n},- for every
*n*, there are*n*consecutive composite Fibonacci numbers, - every positive integer can be written as a sum of distinct Fibonacci numbers, and
- the product of any four consecutive Fibonacci numbers is the area of a Pythagorean triangle.

**See Also:** Fibonacci, FibonacciPrime, WallSunSunPrime, LucasNumber

**Related pages** (outside of this work)

- Fibonacci numbers and the golden section
- Factorizations of Fibonacci and Lucas numbers
- Table of Largest Known Fibonacci primes

Printed from the PrimePages <primes.utm.edu> © Chris Caldwell.