# Definition:Fibonacci Number

## Contents

## Definition

The **Fibonacci numbers** are a sequence $\left \langle {F_n}\right \rangle$ of integers which is formally defined recursively as:

- $F_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n - 1} + F_{n - 2} & : \text{otherwise} \end{cases}$

for all $n \in \Z_{\ge 0}$.

That is, the next integer in the sequence is found by adding together the two previous ones.

### Negative

The definition of **Fibonacci numbers** for negative integers is an extension of the definition for positive integers:

- $F_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n + 2} - F_{n + 1} & : n < 0 \end{cases}$

for all $n \in \Z$.

## Sequence

The sequence of Fibonacci numbers begins:

- $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots$

This sequence is A000045 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Examples

### Fibonacci Number $F_{173}$

The Fibonacci number $F_{173}$ is:

- $638 \, 817 \, 435 \, 613 \, 190 \, 763 \, 972 \, 389 \, 505 \, 493$

It can be factorized as:

- $6 \, 260 \, 874 \, 567 \, 713 \times 102 \, 033 \, 258 \, 884 \, 875 \, 061$

### Fibonacci Number $F_{1000}$

The Fibonacci number $F_{1000}$ is a number with $209$ decimal digits beginning with $4$.

## Also known as

According to some sources, this sequence is also known as Lamé's Sequence, after Gabriel Lamé.

However, this suggestion is difficult to corroborate.

The notation for the $n$th **Fibonacci number** is not universally standardised.

In much professional literature, $u_n$ is used.

$F_n$ tends to appear more in amateur publications, but $F_n$ is rapidly taking over as standard.

$F_n$ is the notation used in *The Fibonacci Quarterly*.

## Also see

- Results about
**Fibonacci numbers**can be found here.

## Source of Name

This entry was named for Leonardo Fibonacci.

## Historical Note

Leonardo Fibonacci famously discussed this sequence in his *Liber Abaci*, in the context of breeding pairs of rabbits.

Hence the name **Fibonacci numbers**, which was given to this sequence by Édouard Lucas, who studied them in detail.

The sequence $\left \langle {F_n}\right \rangle$ was known to Indian mathematicians as long ago as the $7$th century C.E.

It was also studied by Gopāla before $1135$, and by Acharya Hemachandra in about $1150$.

Hence some sources refer to these numbers as the **Gopala-Hemachandra numbers**.

They are also discussed by Johannes Kepler in his work of $1611$ *De Nive Sexangula* (*On the Six-Cornered Snowflake*). It is suspected that Kepler was himself unfamiliar with Fibonacci's work.

Kepler himself had noticed the appearance of Fibonacci numbers in the growth of plants:

*It is in the likeness of this self-developing series that the faculty of propagation is, in my opinion, formed; and so in a flower the authentic flag of this faculty is shown, the pentagon. I pass over all the other arguments that a delightful rumination could adduce in proof of this.*

## Linguistic Note

**Fibonacci** is an Italian surname whose pronunciation is something like ** Fib-bo-nat-chi**, or

**, according to taste.**

*Fib*-bo-*nar*-chiAvoid pronouncing it ** Fie-bo-nac-ky**.

## Sources

- 1957: George Bergman:
*Number System with an Irrational Base*(*Math. Mag.***Vol. 31**,*no. 2*: 98 – 110) www.jstor.org/stable/3029218 - 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {1-1}$ Principle of Mathematical Induction: Exercise $7$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: $(1)$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $4$: Lure of the Unknown

- For a video presentation of the contents of this page, visit the Khan Academy.