# Fibonacci numbers

2010 Mathematics Subject Classification: Primary: 11B39 [MSN][ZBL]

A recursive sequence of integers in which each term is the sum of the two preceding. The elements of the sequence $u_1,u_2,\ldots$ are given by the initial values $u_1 = u_2 = 1$ and the recurrence relation $u_{n+1} = u_n + u_{n-1}$. The first 14 Fibonacci numbers were produced for the first time in 1228 in the manuscripts of Leonardo da Pisa (Fibonacci).

Operations that can be performed on the indices of the Fibonacci numbers can be reduced to operations on the numbers themselves. The basis for this lies in the "addition formula" : $$u_{n+m} = u_{m-1} u_n + u_m u_{n+1}$$

Immediate corollaries of it are: $$u_{2n} = u_n(u_{n+1} + u_{n-1})\,\ \ u_{3n} = u_{n+1}^3 + u_n^3 - u_{n-1}^3$$ etc. The general "multiplication formula" is more complicated: $$u_{mn} = \sum_{k=1}^m \binom{m}{k} u_k u_n^k u_{n-1}^{m-k} \ .$$ The elementary divisibility properties of the Fibonacci numbers are mainly determined by the following facts: $u_{(m,n)} = (u_m,u_n)$; if $p$ is a prime number of the form $p \equiv \pm 1 \pmod 5$, then $u_{p-1}$ is divisible by $p$, while if it is of the form $p \equiv \pm 2 \pmod 5$, then $u_{p+1}$ is; if $u_n$ is divisible by a prime number $q$ and if $p \neq q$, then $u_{pn}/u_n$ is not divisible by $q$; if $u_n$ is divisible by a prime number $p \neq 2$, then $u_{pn}/u_n$ is divisible by $p$, but not by $p^2$; if $u_n$ is divisible by 4, then $u_{2n}/u_n$ is divisible by 2, but not by 4; if $u_n$ is divisible by 2 but not by 4, then $u_{2n}/u_n$ is divisible by 4, but not by 8. At the same time, some number-theoretic problems connected with Fibonacci numbers are extremely hard. For example, the question of whether the set of prime Fibonacci numbers is finite or not has not been solved (1984).

An important role in the theory of Fibonacci numbers is played by the number $\phi = (1+\sqrt 5)/2$, which is a root of the equation $x^2 - x - 1 = 0$. Thus Binet's formula $$u_n = \frac{1}{\sqrt 5}(\phi^n - (-\phi)^{-n})$$ holds; it implies that $u_n$ is the nearest integer to $\phi^n/\sqrt 5$, and that $$\lim_{n \rightarrow \infty} \frac{u_{n+1}}{u_n} = \phi \ .$$

The Fibonacci numbers occupy a special position in the theory of continued fractions. In the continued-fraction expansion of $u_{n+1}/u_n$ all the partial quotients are ones and the number of them is not less than that of the incomplete quotients of the expansion of any other fraction with denominator less than $u_n$. In a certain sense the number $\phi$ is described by its approximating fractions $u_{n+1}/u_n$ in a "worst possible" way.

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How to Cite This Entry:
Fibonacci numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fibonacci_numbers&oldid=36001
This article was adapted from an original article by N.N. Vorob'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article