Fibonacci number
Overview
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Fibonacci numbers are the numbers in the following integer sequence
Integer sequence
In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
: .
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.
In mathematical terms, the sequence F_{n} of Fibonacci numbers is defined by the recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
with seed values
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci.
Discussions
Encyclopedia
In mathematics
, the Fibonacci numbers are the numbers in the following integer sequence
: .
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.
In mathematical terms, the sequence F_{n} of Fibonacci numbers is defined by the recurrence relation
with seed values
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci's 1202 book Liber Abaci
introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics
.
(By modern convention, the sequence begins with F_{0} = 0. The Liber Abaci began the sequence with F_{1} = 1, omitting the initial 0, and the sequence is still written this way by some.)
Fibonacci numbers are closely related to Lucas number
s in that they are a complementary pair of Lucas sequence
s. They are intimately connected with the golden ratio
, for example the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... . Applications include computer algorithms such as the Fibonacci search technique
and the Fibonacci heap
data structure, and graphs called Fibonacci cube
s used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, arrangement of leaves on a stem
, the fruit spouts of a pineapple
, the flowering of artichoke
, an uncurling fern
and the arrangement of a pine cone.
, in connection with Sanskrit prosody
. In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with the short (S), and counting the different patterns of L and S within a given fixed length results in the Fibonacci numbers; the number of patterns that are m short syllables long is the Fibonacci number F_{m + 1}.
Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in part to Pingala
(200 BC), later being associated with Virahanka
(c. 700 AD), Gopāla (c.1135 AD), and Hemachandra (c.1150)". Parmanand Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and cites scholars who interpret it in context as saying that the cases for m beats (F_{m+1}) is obtained by adding a [S] to F_{m} cases and [L] to the F_{m−1} cases. He dates Pingala before 450 BCE.
However, the clearest exposition of the series arises in the work of Virahanka
(c. 700AD), whose own work is lost, but is available in a quotation by Gopala (c.1135):
The series is also discussed by Gopala (before 1135AD) and by the Jain scholar Hemachandra (c. 1150AD).
In the West, the Fibonacci sequence first appears in the book Liber Abaci
(1202) by Leonardo of Pisa, known as Fibonacci
. Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit
population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.
The name "Fibonacci sequence" was first used by the 19thcentury number theorist Édouard Lucas
.
The sequence can also be extended to negative index n using the rearranged recurrence relation
which yields the sequence of "negafibonacci" numbers satisfying
Thus the complete sequence is
(see Binomial coefficient
).
The Fibonacci numbers can be found in different ways in the sequence of binary
strings
.
's formula, even though it was already known by Abraham de Moivre
:
where
is the golden ratio
, and
To see this, note that φ and ψ are both solutions of the equations
so the powers of φ and ψ satisfy the Fibonacci recursion. In other words
and
It follows that for any values a and b, the sequence defined by
satisfies the same recurrence
If a and b are chosen so that U_{0} = 0 and U_{1} = 1 then the resulting sequence U_{n} must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:
which has solution
producing the required formula.
for all n ≥ 0, the number F_{n} is the closest integer to
Therefore it can be found by rounding, or in terms of the floor function
:
Similarly, if we already know that the number F > 1 is a Fibonacci number, we can determine its index within the sequence by
observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that the limit approaches the golden ratio .
This convergence does not depend on the starting values chosen, excluding 0, 0. For example, the initial values 19 and 31 generate the sequence 19, 31, 50, 81, 131, 212, 343, 555 ... etc. The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
In fact this holds for any sequence which satisfies the Fibonacci recurrence other than a sequence of 0's. This can be derived from Binet's formula.
this expression can be used to decompose higher powers as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of and 1. The resulting recurrence relation
ships yield Fibonacci numbers as the linear coefficients:
This expression is also true for if the Fibonacci sequence is extended to negative integers using the Fibonacci rule
The eigenvalues of the matrix A are and , and the elements of the eigenvectors of A, and , are in the ratios and Using these facts, and the properties of eigenvalues, we can derive a direct formula for the nth element in the Fibonacci series:
The matrix has a determinant
of −1, and thus it is a 2×2 unimodular matrix
. This property can be understood in terms of the continued fraction
representation for the golden ratio:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The matrix representation gives the following closed expression for the Fibonacci numbers:
Taking the determinant of both sides of this equation yields Cassini's identity
Additionally, since for any square matrix A, the following identities can be derived:
In particular, with ,
which is true if and only if
z is a Fibonacci number. In this formula, can be computed rapidly using any of the previously discussed closedform expressions.
One implication of the above expression is this: if it is known that a number z is a Fibonacci number, we may determine an n such that F(n) = z by the following:
Alternatively, a positive integer z is a Fibonacci number if and only if one of or is a perfect square
.
A slightly more sophisticated test uses the fact that the convergent
s of the continued fraction
representation of are ratios of successive Fibonacci numbers. That is, the inequality
(with coprime
positive integers p, q) is true if and only if p and q are successive Fibonacci numbers. From this one derives the criterion that z is a Fibonacci number if and only if the closed interval
contains a positive integer. For , it is easy to show that this interval contains at most one integer, and in the event that z is a Fibonacci number, the contained integer is equal to the next successive Fibonacci number after z. Somewhat remarkably, this result still holds for the case , but it must be stated carefully since appears twice in the Fibonacci sequence, and thus has two distinct successors.
.
F(n) can be interpreted as the number of sequences of 1s and 2s that sum to n − 1, with the convention that F(0) = 0, meaning no sum will add up to −1, and that F(1) = 1, meaning the empty sum will "add up" to 0.
Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice.
The sum of the first n − 1 Fibonacci numbers, F_{j}, such that j is odd, is the (2n)th Fibonacci number.
The sum of the first n Fibonacci numbers, F_{j}, such that j is even, is the (2n + 1)th Fibonacci number minus 1.
Where is the nth Lucas Number.
from which other identities for specific values of k, n, and c can be derived below, including
for all integers n and k. Doubling identities of this type can be used to calculate F_{n} using O(log n) long multiplication operations of size n bits. The number of bits of precision needed to perform each multiplication doubles at each step, so the performance is limited by the final multiplication; if the fast Schönhage–Strassen multiplication algorithm is used, this is O(n log n log log n) bit operations. Notice that, with the definition of Fibonacci numbers with negative n given in the introduction, this formula reduces to the double n formula when k = 0.
s, which have the same recursive properties but start with L_{0} = 2 and L_{1} = 1. These properties include F_{2n} = F_{n}L_{n}.
There are also scaling identities, which take you from F_{n} and F_{n+1} to a variety of things of the form F_{an+b}; for instance
These can be found experimentally using lattice reduction
, and are useful in setting up the special number field sieve
to factorize
a Fibonacci number. Such relations exist in a very general sense for numbers defined by recurrence relations. See the section on multiplication formulae under Perrin number
s for details.
of the Fibonacci sequence is the power series
This series has a simple and interesting closedform solution for :
This solution can be proven by using the Fibonacci recurrence to expand each coefficient in the infinite sum defining :
Solving the equation for results in the closed form solution.
In particular, math puzzlebooks note the curious value , or more generally
for all integers .
Conversely,
and the sum of squared reciprocal Fibonacci numbers as
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
and there is a nice nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio
,
Results such as these make it plausible that a closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but none is yet known. Despite that, the reciprocal Fibonacci constant
has been proved irrational
by Richard AndréJeannin.
Millin series gives a remarkable identity:
which follows from the closed form for its partial sums as N tends to infinity:
. The first few are:
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.
F_{kn} is divisible by F_{n}, so, apart from F_{4} = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long
runs of composite number
s, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
With the exceptions of 1, 8 and 144 (F_{1} = F_{2}, F_{6} and F_{12}) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem
).
144 is the only nontrivial
square
Fibonacci number. Attila Pethő proved in 2001 that there are only finitely many perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that only 8 and 144 are nontrivial perfect powers.
No Fibonacci number greater than F_{6} = 8 is one greater or one less than a prime number.
Any three consecutive Fibonacci numbers, taken two at a time, are relatively prime: that is,
More generally,
which is evaluated as follows:
If p is a prime number then
It is not known whether there exists a prime p such that . Such primes (if there are any) would be called Wall–Sun–Sun primes.
Also, if p ≠ 5 is an odd prime number then:
For odd n, all odd prime divisors of F_{n} are ≡ 1 (mod 4), implying that all odd divisors of F_{n} (as the products of odd prime divisors) are ≡ 1 (mod 4).
with period at most n^{2}1. The lengths of the periods for various n form the socalled Pisano period
s . Determining the Pisano periods in general is an open problem, although for any particular n it can be solved as an instance of cycle detection.
. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.
The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely. The triangle sides a, b, c can be calculated directly:
These formulas satisfy for all n, but they only represent triangle sides when n > 2.
Any four consecutive Fibonacci numbers F_{n}, F_{n+1}, F_{n+2} and F_{n+3} can also be used to generate a Pythagorean triple in a different way:
Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then:
to , the number of digits in is asymptotic to . As a consequence, for every integer there are either 4 or 5 Fibonacci numbers with d decimal digits.
More generally, in the base b representation, the number of digits in is asymptotic to .
to determine the greatest common divisor
of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
Yuri Matiyasevich
was able to show that the Fibonacci numbers can be defined by a Diophantine equation
, which led to his original solution of Hilbert's tenth problem
.
The Fibonacci numbers are also an example of a complete sequence
. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most. Specifically, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem
, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding
.
Fibonacci numbers are used by some pseudorandom number generators.
Fibonacci numbers are used in a polyphase version of the merge sort
algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ. A tapedrive implementation of the polyphase merge sort
was described in The Art of Computer Programming
.
Fibonacci numbers arise in the analysis of the Fibonacci heap
data structure.
The Fibonacci cube
is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology
for parallel computing
.
A onedimensional optimization method, called the Fibonacci search technique
, uses Fibonacci numbers.
The Fibonacci number series is used for optional lossy compression in the IFF
8SVX
audio file format used on Amiga
computers. The number series compands
the original audio wave similar to logarithmic methods such as µlaw.
In music
, Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements. It is commonly thought that the third movement of Béla Bartók
's Music for Strings, Percussion, and Celesta was structured using Fibonacci numbers.
Since the conversion
factor 1.609344 for miles to kilometers is close to the golden ratio (denoted φ), the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix
2 number register
in golden ratio base
φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.
, the fruitlets of a pineapple
, the flowering of artichoke
, an uncurling fern and the arrangement of a pine cone. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g., relating to the breeding of rabbits, the spirals of shells, and the curve of waves. The Fibonacci numbers are also found in the family tree of honeybees.
Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free group
s, specifically as certain Lindenmayer grammar
s.
A model for the pattern of florets in the head of a sunflower
was proposed by H. Vogel in 1979.
This has the form
where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle
, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j which depends on r, the distance from the center. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.
Thus, a male bee will always have one parent, and a female bee will have two.
If one traces the ancestry of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 greatgrandparents, 5 greatgreatgrandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, F_{n}, is the number of female ancestors, which is F_{n−1}, plus the number of male ancestors, which is F_{n−2}. (This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Fibonacci numbers are the numbers in the following integer sequence
Integer sequence
In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
: .
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.
In mathematical terms, the sequence F_{n} of Fibonacci numbers is defined by the recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
with seed values
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci's 1202 book Liber Abaci
Liber Abaci
Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci...
introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
.
(By modern convention, the sequence begins with F_{0} = 0. The Liber Abaci began the sequence with F_{1} = 1, omitting the initial 0, and the sequence is still written this way by some.)
Fibonacci numbers are closely related to Lucas number
Lucas number
The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...
s in that they are a complementary pair of Lucas sequence
Lucas sequence
In mathematics, the Lucas sequences Un and Vn are certain integer sequences that satisfy the recurrence relationwhere P and Q are fixed integers...
s. They are intimately connected with the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
, for example the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... . Applications include computer algorithms such as the Fibonacci search technique
Fibonacci search technique
In computer science, the Fibonacci search technique is a method of searching a sorted array using a divide and conquer algorithm that narrows down possible locations with the aid of Fibonacci numbers.Compared to binary search, Fibonacci search examines...
and the Fibonacci heap
Fibonacci heap
In computer science, a Fibonacci heap is a heap data structure consisting of a collection of trees. It has a better amortized running time than a binomial heap. Fibonacci heaps were developed by Michael L. Fredman and Robert E. Tarjan in 1984 and first published in a scientific journal in 1987...
data structure, and graphs called Fibonacci cube
Fibonacci cube
The Fibonacci cubes or Fibonacci networks are a family of undirected graphs with rich recursive properties derived from its origin in Number Theory. Mathematically they are similar to the hypercube graphs, but with a Fibonacci number of vertices, studied in graphtheoretic mathematics...
s used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, arrangement of leaves on a stem
Phyllotaxis
In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem . Pattern structure :...
, the fruit spouts of a pineapple
Pineapple
Pineapple is the common name for a tropical plant and its edible fruit, which is actually a multiple fruit consisting of coalesced berries. It was given the name pineapple due to its resemblance to a pine cone. The pineapple is by far the most economically important plant in the Bromeliaceae...
, the flowering of artichoke
Artichoke
Plants:* Globe artichoke, a partially edible perennial thistle originating in southern Europe around the Mediterranean* Jerusalem artichoke, a species of sunflower with an edible tuber...
, an uncurling fern
Fern
A fern is any one of a group of about 12,000 species of plants belonging to the botanical group known as Pteridophyta. Unlike mosses, they have xylem and phloem . They have stems, leaves, and roots like other vascular plants...
and the arrangement of a pine cone.
Origins
The Fibonacci sequence appears in Indian mathematicsIndian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
, in connection with Sanskrit prosody
Sanskrit prosody
Versification in Classical Sanskrit poetry is of three kinds.# Syllabic verse : meters depend on the number of syllables in a verse, with relative freedom in the distribution of light and heavy syllables...
. In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with the short (S), and counting the different patterns of L and S within a given fixed length results in the Fibonacci numbers; the number of patterns that are m short syllables long is the Fibonacci number F_{m + 1}.
Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in part to Pingala
Pingala
Pingala is the traditional name of the author of the ' , the earliest known Sanskrit treatise on prosody.Nothing is known about Piṅgala himself...
(200 BC), later being associated with Virahanka
Virahanka
Virahanka was an Indian prosodist who is also known for his work on mathematics. He possibly lived in the 6th century, but it is also possible that this date may be as late as 8th century.His work on prosody builds on the Chhandasutras of...
(c. 700 AD), Gopāla (c.1135 AD), and Hemachandra (c.1150)". Parmanand Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and cites scholars who interpret it in context as saying that the cases for m beats (F_{m+1}) is obtained by adding a [S] to F_{m} cases and [L] to the F_{m−1} cases. He dates Pingala before 450 BCE.
However, the clearest exposition of the series arises in the work of Virahanka
Virahanka
Virahanka was an Indian prosodist who is also known for his work on mathematics. He possibly lived in the 6th century, but it is also possible that this date may be as late as 8th century.His work on prosody builds on the Chhandasutras of...
(c. 700AD), whose own work is lost, but is available in a quotation by Gopala (c.1135):
 Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mAtrAvr.ttas (prosodic combinations).
The series is also discussed by Gopala (before 1135AD) and by the Jain scholar Hemachandra (c. 1150AD).
In the West, the Fibonacci sequence first appears in the book Liber Abaci
Liber Abaci
Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci...
(1202) by Leonardo of Pisa, known as Fibonacci
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...
. Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit
Rabbit
Rabbits are small mammals in the family Leporidae of the order Lagomorpha, found in several parts of the world...
population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
 At the end of the first month, they mate, but there is still only 1 pair.
 At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
 At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
 At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.
The name "Fibonacci sequence" was first used by the 19thcentury number theorist Édouard Lucas
Edouard Lucas
François Édouard Anatole Lucas was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.Biography:...
.
List of Fibonacci numbers
The first 21 Fibonacci numbers F_{n} for n = 0, 1, 2, ..., 20 are:F_{0}  F_{1}  F_{2}  F_{3}  F_{4}  F_{5}  F_{6}  F_{7}  F_{8}  F_{9}  F_{10}  F_{11}  F_{12}  F_{13}  F_{14}  F_{15}  F_{16}  F_{17}  F_{18}  F_{19}  F_{20} 
0  1  1  2  3  5  8  13  21  34  55  89  144  233  377  610  987  1597  2584  4181  6765 
The sequence can also be extended to negative index n using the rearranged recurrence relation
which yields the sequence of "negafibonacci" numbers satisfying
Thus the complete sequence is
F_{−8}  F_{−7}  F_{−6}  F_{−5}  F_{−4}  F_{−3}  F_{−2}  F_{−1}  F_{0}  F_{1}  F_{2}  F_{3}  F_{4}  F_{5}  F_{6}  F_{7}  F_{8} 
−21  13  −8  5  −3  2  −1  1  0  1  1  2  3  5  8  13  21 
Occurrences in mathematics
The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's trianglePascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
(see Binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
).
The Fibonacci numbers can be found in different ways in the sequence of binary
Binary numeral system
The binary numeral system, or base2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base2 system is a positional notation with a radix of 2...
strings
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
.
 The number of binary strings of length n without consecutive 1s is the Fibonacci number F_{n+2}. For example, out of the 16 binary strings of length 4, there are F_{6} = 8 without consecutive 1s – they are 0000, 0100, 0010, 0001, 0101, 1000, 1010 and 1001. By symmetry, the number of strings of length n without consecutive 0s is also F_{n+2}.
 The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number F_{n+1}. For example, out of the 16 binary strings of length 4, there are F_{5} = 5 without an odd number of consecutive 1s – they are 0000, 0011, 0110, 1100, 1111.
 The number of binary strings of length n without an even number of consecutive 0s or 1s is 2F_{n}. For example, out of the 16 binary strings of length 4, there are 2F_{4} = 6 without an even number of consecutive 0s or 1s – they are 0001, 1000, 1110, 0111, 0101, 1010.
Closedform expression
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closedform solution. It has become known as BinetJacques Philippe Marie Binet
Jacques Philippe Marie Binet was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical foundations of matrix algebra which would later lead to important contributions by Cayley...
's formula, even though it was already known by Abraham de Moivre
Abraham de Moivre
Abraham de Moivre was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling...
:
where
is the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
, and
To see this, note that φ and ψ are both solutions of the equations
so the powers of φ and ψ satisfy the Fibonacci recursion. In other words
and
It follows that for any values a and b, the sequence defined by
satisfies the same recurrence
If a and b are chosen so that U_{0} = 0 and U_{1} = 1 then the resulting sequence U_{n} must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:
which has solution
producing the required formula.
Computation by rounding
Sincefor all n ≥ 0, the number F_{n} is the closest integer to
Therefore it can be found by rounding, or in terms of the floor function
Floor function
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...
:
Similarly, if we already know that the number F > 1 is a Fibonacci number, we can determine its index within the sequence by
Limit of consecutive quotients
Johannes KeplerJohannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that the limit approaches the golden ratio .
This convergence does not depend on the starting values chosen, excluding 0, 0. For example, the initial values 19 and 31 generate the sequence 19, 31, 50, 81, 131, 212, 343, 555 ... etc. The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
In fact this holds for any sequence which satisfies the Fibonacci recurrence other than a sequence of 0's. This can be derived from Binet's formula.
Decomposition of powers of the golden ratio
Since the golden ratio satisfies the equationthis expression can be used to decompose higher powers as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of and 1. The resulting recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
ships yield Fibonacci numbers as the linear coefficients:
This expression is also true for if the Fibonacci sequence is extended to negative integers using the Fibonacci rule
Matrix form
A 2dimensional system of linear difference equations that describes the Fibonacci sequence isThe eigenvalues of the matrix A are and , and the elements of the eigenvectors of A, and , are in the ratios and Using these facts, and the properties of eigenvalues, we can derive a direct formula for the nth element in the Fibonacci series:
The matrix has a determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of −1, and thus it is a 2×2 unimodular matrix
Unimodular matrix
In mathematics, a unimodular matrix M is a square integer matrix with determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N which is its inverse...
. This property can be understood in terms of the continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
representation for the golden ratio:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The matrix representation gives the following closed expression for the Fibonacci numbers:
Taking the determinant of both sides of this equation yields Cassini's identity
Additionally, since for any square matrix A, the following identities can be derived:
In particular, with ,
Recognizing Fibonacci numbers
The question may arise whether a positive integer z is a Fibonacci number. Since is the closest integer to , the most straightforward, bruteforce test is the identitywhich is true if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
z is a Fibonacci number. In this formula, can be computed rapidly using any of the previously discussed closedform expressions.
One implication of the above expression is this: if it is known that a number z is a Fibonacci number, we may determine an n such that F(n) = z by the following:
Alternatively, a positive integer z is a Fibonacci number if and only if one of or is a perfect square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
.
A slightly more sophisticated test uses the fact that the convergent
Convergent (continued fraction)
A convergent is one of a sequence of values obtained by evaluating successive truncations of a continued fraction The nth convergent is also known as the nth approximant of a continued fraction.Representation of real numbers:...
s of the continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
representation of are ratios of successive Fibonacci numbers. That is, the inequality
(with coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
positive integers p, q) is true if and only if p and q are successive Fibonacci numbers. From this one derives the criterion that z is a Fibonacci number if and only if the closed interval
contains a positive integer. For , it is easy to show that this interval contains at most one integer, and in the event that z is a Fibonacci number, the contained integer is equal to the next successive Fibonacci number after z. Somewhat remarkably, this result still holds for the case , but it must be stated carefully since appears twice in the Fibonacci sequence, and thus has two distinct successors.
Identities
Most identities involving Fibonacci numbers draw from combinatorial argumentsCombinatorial proof
In mathematics, the term combinatorial proof is often used to mean either of two types of proof of an identity in enumerative combinatorics that either states that two sets of combinatorial configurations, depending on one or more parameters, have the same number of elements , or gives a formula...
.
F(n) can be interpreted as the number of sequences of 1s and 2s that sum to n − 1, with the convention that F(0) = 0, meaning no sum will add up to −1, and that F(1) = 1, meaning the empty sum will "add up" to 0.
Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice.
First identity
 For n > 1.
 The nth Fibonacci number is the sum of the previous two Fibonacci numbers.
Proof 

We must establish that the sequence of numbers defined by the combinatorial interpretation above satisfy the same recurrence relation as the Fibonacci numbers (and so are indeed identical to the Fibonacci numbers). The set of F(n + 1) ways of making ordered sums of 1s and 2s that sum to n may be divided into two nonoverlapping sets. The first set contains those sums whose first summand is 1; the remainder sums to n − 1, so there are F(n) sums in the first set. The second set contains those sums whose first summand is 2; the remainder sums to n − 2, so there are F(n − 1) sums in the second set. The first summand can only be 1 or 2, so these two sets exhaust the original set. Thus F(n + 1) = F(n) + F(n − 1). 
Second identity
The sum of the first n Fibonacci numbers is equal to the n+2nd Fibonacci number minus 1. In symbols: The sum of the first n Fibonacci numbers is the (n + 2)nd Fibonacci number minus 1.
Proof 

We count the number of ways summing 1s and 2s to n + 1 such that at least one of the summands is 2. As before, there are F(n + 2) ways summing 1s and 2s to n + 1 when n ≥ 0. Since there is only one sum of n + 1 that does not use any 2, namely 1 + ... + 1 (n + 1 terms), we subtract 1 from F(n + 2). Equivalently, we can consider the first occurrence of 2 as a summand. If, in a sum, the first summand is 2, then there are F(n) ways to the complete the counting for n − 1. If the second summand is 2 but the first is 1, then there are F(n − 1) ways to complete the counting for n − 2. Proceed in this fashion. Eventually we consider the (n + 1)th summand. If it is 2 but all of the previous n summands are 1s, then there are F(0) ways to complete the counting for 0. If a sum contains 2 as a summand, the first occurrence of such summand must take place in between the first and (n + 1)th position. Thus F(n) + F(n − 1) + ... + F(0) gives the desired counting. By induction:

Third identity
This identity has slightly different forms for F_{j}, depending on whether j is odd or even.The sum of the first n − 1 Fibonacci numbers, F_{j}, such that j is odd, is the (2n)th Fibonacci number.
The sum of the first n Fibonacci numbers, F_{j}, such that j is even, is the (2n + 1)th Fibonacci number minus 1.
Proofs 

1: j is odd By induction for F_{2n}: A basis case for this could be F_{1} = F_{2}. 2: j is even By induction for F_{2n+1}: A basis case for this could be F_{0} = F_{1} − 1. 
Alternative proof 

By using identity 1 we can construct a telescoping sum: If the summands are the Fibonacci numbers with even index, the proof is very similar. Summing both cases yields identity 2. 
Fourth identity
Proof 

This identity can be established in two stages. First, we count the number of ways summing 1s and 2s to −1, 0, ..., or n + 1 such that at least one of the summands is 2. By our second identity, there are F(n + 2) − 1 ways summing to n + 1; F(n + 1) − 1 ways summing to n; ...; and, eventually, F(2) − 1 way summing to 1. As F(1) − 1 = F(0) = 0, we can add up all n + 1 sums and apply the second identity again to obtain
On the other hand, we observe from the second identity that there are
......
Adding up all n + 1 sums, we see that there are
Since the two methods of counting refer to the same number, we have
Finally, we complete the proof by subtracting the above identity from n + 1 times the second identity. 
Fifth identity
 The sum of the squares of the first n Fibonacci numbers is the product of the nth and (n + 1)th Fibonacci numbers.
Proof 

Although this identity can be established by either induction or direct, albeit messy, algebraic manipulation, perhaps the most elegant and most insightful method is by a simple geometric argument. Consider the Fibonacci Rectangles constructed in previous sections. Using a common trick, we will compute the area of this rectangle in two different ways. But since this must yield the same answer in both cases, we know these resulting expressions must be equal, which will yield the desired identity. On the one hand, the nth rectangle is composed of n squares, whose side lengths are F(1), F(2), ..., F(n). Its area is therefore the sum of each of these squares, which is given by On the other hand, we know that the nth rectangle has side lengths F(n) and F(n + 1). Thus, its area is simply given by Setting these expressions equal to each other completes the proof. 
Identity for doubling n
Where is the nth Lucas Number.
Another identity
Another identity useful for calculating F_{n} for large values of n isfrom which other identities for specific values of k, n, and c can be derived below, including
for all integers n and k. Doubling identities of this type can be used to calculate F_{n} using O(log n) long multiplication operations of size n bits. The number of bits of precision needed to perform each multiplication doubles at each step, so the performance is limited by the final multiplication; if the fast Schönhage–Strassen multiplication algorithm is used, this is O(n log n log log n) bit operations. Notice that, with the definition of Fibonacci numbers with negative n given in the introduction, this formula reduces to the double n formula when k = 0.
Other identities
Other identities include relationships to the Lucas numberLucas number
The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...
s, which have the same recursive properties but start with L_{0} = 2 and L_{1} = 1. These properties include F_{2n} = F_{n}L_{n}.
There are also scaling identities, which take you from F_{n} and F_{n+1} to a variety of things of the form F_{an+b}; for instance
 by Cassini's identity.
These can be found experimentally using lattice reduction
Lattice reduction
In mathematics, the goal of lattice basis reduction is given an integer lattice basis as input, to find a basis with short, nearly orthogonal vectors. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice.Nearly...
, and are useful in setting up the special number field sieve
Special number field sieve
In number theory, a branch of mathematics, the special number field sieve is a specialpurpose integer factorization algorithm. The general number field sieve was derived from it....
to factorize
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
a Fibonacci number. Such relations exist in a very general sense for numbers defined by recurrence relations. See the section on multiplication formulae under Perrin number
Perrin number
In mathematics, the Perrin numbers are defined by the recurrence relationandThe sequence of Perrin numbers starts withThe number of different maximal independent sets in an nvertex cycle graph is counted by the nth Perrin number.History:...
s for details.
Power series
The generating functionGenerating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
of the Fibonacci sequence is the power series
This series has a simple and interesting closedform solution for :
This solution can be proven by using the Fibonacci recurrence to expand each coefficient in the infinite sum defining :
Solving the equation for results in the closed form solution.
In particular, math puzzlebooks note the curious value , or more generally
for all integers .
Conversely,
Reciprocal sums
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every oddindexed reciprocal Fibonacci number asand the sum of squared reciprocal Fibonacci numbers as
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
and there is a nice nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
,
Results such as these make it plausible that a closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but none is yet known. Despite that, the reciprocal Fibonacci constant
has been proved irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b nonzero, and is therefore not a rational number....
by Richard AndréJeannin.
Millin series gives a remarkable identity:
which follows from the closed form for its partial sums as N tends to infinity:
Divisibility properties
Every 3rd number of the sequence is even and more generally, every kth number of the sequence is a multiple of F_{k}. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility propertyFibonacci primes
A Fibonacci prime is a Fibonacci number that is primePrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. The first few are:
 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, … .
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.
F_{kn} is divisible by F_{n}, so, apart from F_{4} = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long
Arbitrarily large
In mathematics, the phrase arbitrarily large, arbitrarily small, arbitrarily long is used in statements such as:which is shorthand for:This should not be confused with the phrase "sufficiently large"...
runs of composite number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
s, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
With the exceptions of 1, 8 and 144 (F_{1} = F_{2}, F_{6} and F_{12}) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem
Carmichael's theorem
Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that for n greater than 12, the nth Fibonacci number F has at least one prime factor that is not a factor of any earlier Fibonacci number. The only exceptions for n up to 12 are:Carmichael's theorem, named after...
).
144 is the only nontrivial
square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
Fibonacci number. Attila Pethő proved in 2001 that there are only finitely many perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that only 8 and 144 are nontrivial perfect powers.
No Fibonacci number greater than F_{6} = 8 is one greater or one less than a prime number.
Any three consecutive Fibonacci numbers, taken two at a time, are relatively prime: that is,
 gcdGreatest common divisorIn mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more nonzero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
(F_{n}, F_{n+1}) = gcd(F_{n}, F_{n+2}) = 1.
More generally,
 gcd(F_{n}, F_{m}) = F_{gcd(n, m).}
Prime divisors of Fibonacci numbers
The divisibility of Fibonacci numbers by a prime p is related to the Legendre symbolLegendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic nonresidue is −1....
which is evaluated as follows:
If p is a prime number then
For example,
It is not known whether there exists a prime p such that . Such primes (if there are any) would be called Wall–Sun–Sun primes.
Also, if p ≠ 5 is an odd prime number then:
Examples of all the cases:
For odd n, all odd prime divisors of F_{n} are ≡ 1 (mod 4), implying that all odd divisors of F_{n} (as the products of odd prime divisors) are ≡ 1 (mod 4).
For example,
F_{1} = 1, F_{3} = 2, F_{5} = 5, F_{7} = 13, F_{9} = 34 = 2×17, F_{11} = 89, F_{13} = 233, F_{15} = 610 = 2×5×61
Periodicity modulo n
It may be seen that if the members of the Fibonacci sequence are taken mod n, the resulting sequence must be periodicPeriodic sequence
In mathematics, a periodic sequence is a sequence for which the same terms are repeated over and over:The number p of repeated terms is called the period.Definition:A periodic sequence is a sequence a1, a2, a3, ... satisfying...
with period at most n^{2}1. The lengths of the periods for various n form the socalled Pisano period
Pisano period
In number theory, the nth Pisano period, written π, is the period with which the sequence of Fibonacci numbers, modulo n repeats. For example, the Fibonacci numbers modulo 3 are , 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, etc., with the first eight numbers repeating, so π = 8.Pisano periods are named after...
s . Determining the Pisano periods in general is an open problem, although for any particular n it can be solved as an instance of cycle detection.
Right triangles
Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triplePythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a wellknown example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...
. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.
The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely. The triangle sides a, b, c can be calculated directly:
These formulas satisfy for all n, but they only represent triangle sides when n > 2.
Any four consecutive Fibonacci numbers F_{n}, F_{n+1}, F_{n+2} and F_{n+3} can also be used to generate a Pythagorean triple in a different way:
Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then:
Magnitude
Since is asymptoticAsymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
to , the number of digits in is asymptotic to . As a consequence, for every integer there are either 4 or 5 Fibonacci numbers with d decimal digits.
More generally, in the base b representation, the number of digits in is asymptotic to .
Applications
The Fibonacci numbers are important in the computational runtime analysis of Euclid's algorithmEuclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
to determine the greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more nonzero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
Yuri Matiyasevich
Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich, is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's tenth problem, presented in his doctoral thesis, at LOMI . Biography :* In 19621963 studied at Saint Petersburg Lyceum 239...
was able to show that the Fibonacci numbers can be defined by a Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
, which led to his original solution of Hilbert's tenth problem
Hilbert's tenth problem
Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900. Its statement is as follows:Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite...
.
The Fibonacci numbers are also an example of a complete sequence
Complete sequence
In mathematics, an integer sequence is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once....
. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most. Specifically, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem
Zeckendorf's theorem
Zeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers....
, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding
Fibonacci coding
In mathematics, Fibonacci coding is a universal code which encodes positive integers into binary code words. Each code word ends with "11" and contains no other instances of "11" before the end.Definition:...
.
Fibonacci numbers are used by some pseudorandom number generators.
Fibonacci numbers are used in a polyphase version of the merge sort
Merge sort
Merge sort is an O comparisonbased sorting algorithm. Most implementations produce a stable sort, meaning that the implementation preserves the input order of equal elements in the sorted output. It is a divide and conquer algorithm...
algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ. A tapedrive implementation of the polyphase merge sort
Polyphase merge sort
A polyphase merge sort is an algorithm which decreases the number of runs at every iteration of the main loop by merging runs into larger runs. It is used for external sorting. Ordinary merge sort :...
was described in The Art of Computer Programming
The Art of Computer Programming
The Art of Computer Programming is a comprehensive monograph written by Donald Knuth that covers many kinds of programming algorithms and their analysis....
.
Fibonacci numbers arise in the analysis of the Fibonacci heap
Fibonacci heap
In computer science, a Fibonacci heap is a heap data structure consisting of a collection of trees. It has a better amortized running time than a binomial heap. Fibonacci heaps were developed by Michael L. Fredman and Robert E. Tarjan in 1984 and first published in a scientific journal in 1987...
data structure.
The Fibonacci cube
Fibonacci cube
The Fibonacci cubes or Fibonacci networks are a family of undirected graphs with rich recursive properties derived from its origin in Number Theory. Mathematically they are similar to the hypercube graphs, but with a Fibonacci number of vertices, studied in graphtheoretic mathematics...
is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology
Network topology
Network topology is the layout pattern of interconnections of the various elements of a computer or biological network....
for parallel computing
Parallel computing
Parallel computing is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently . There are several different forms of parallel computing: bitlevel,...
.
A onedimensional optimization method, called the Fibonacci search technique
Fibonacci search technique
In computer science, the Fibonacci search technique is a method of searching a sorted array using a divide and conquer algorithm that narrows down possible locations with the aid of Fibonacci numbers.Compared to binary search, Fibonacci search examines...
, uses Fibonacci numbers.
The Fibonacci number series is used for optional lossy compression in the IFF
Interchange File Format
Interchange File Format , is a generic container file format originally introduced by the Electronic Arts company in 1985 in order to ease transfer of data between software produced by different companies....
8SVX
8SVX
8SVX is a subformat of the Interchange File Format. The subformat is for 8bit sampled sounds, supports both mono and stereo streams as well as loops; commonly used as a basic audio sample format on Amiga computers for many years...
audio file format used on Amiga
Amiga
The Amiga is a family of personal computers that was sold by Commodore in the 1980s and 1990s. The first model was launched in 1985 as a highend home computer and became popular for its graphical, audio and multitasking abilities...
computers. The number series compands
Companding
In telecommunication, signal processing, and thermodynamics, companding is a method of mitigating the detrimental effects of a channel with limited dynamic range...
the original audio wave similar to logarithmic methods such as µlaw.
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements. It is commonly thought that the third movement of Béla Bartók
Béla Bartók
Béla Viktor János Bartók was a Hungarian composer and pianist. He is considered one of the most important composers of the 20th century and is regarded, along with Liszt, as Hungary's greatest composer...
's Music for Strings, Percussion, and Celesta was structured using Fibonacci numbers.
Since the conversion
Conversion of units
Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors. Process :...
factor 1.609344 for miles to kilometers is close to the golden ratio (denoted φ), the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
2 number register
Processor register
In computer architecture, a processor register is a small amount of storage available as part of a CPU or other digital processor. Such registers are addressed by mechanisms other than main memory and can be accessed more quickly...
in golden ratio base
Golden ratio base
Golden ratio base is a noninteger positional numeral system that uses the golden ratio as its base. It is sometimes referred to as baseφ, golden mean base, phibase, or, colloquially, phinary...
φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.
In nature
Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stemPhyllotaxis
In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem . Pattern structure :...
, the fruitlets of a pineapple
Pineapple
Pineapple is the common name for a tropical plant and its edible fruit, which is actually a multiple fruit consisting of coalesced berries. It was given the name pineapple due to its resemblance to a pine cone. The pineapple is by far the most economically important plant in the Bromeliaceae...
, the flowering of artichoke
Artichoke
Plants:* Globe artichoke, a partially edible perennial thistle originating in southern Europe around the Mediterranean* Jerusalem artichoke, a species of sunflower with an edible tuber...
, an uncurling fern and the arrangement of a pine cone. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g., relating to the breeding of rabbits, the spirals of shells, and the curve of waves. The Fibonacci numbers are also found in the family tree of honeybees.
Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s, specifically as certain Lindenmayer grammar
Lsystem
An Lsystem or Lindenmayer system is a parallel rewriting system, namely a variant of a formal grammar, most famously used to model the growth processes of plant development, but also able to model the morphology of a variety of organisms...
s.
A model for the pattern of florets in the head of a sunflower
Sunflower
Sunflower is an annual plant native to the Americas. It possesses a large inflorescence . The sunflower got its name from its huge, fiery blooms, whose shape and image is often used to depict the sun. The sunflower has a rough, hairy stem, broad, coarsely toothed, rough leaves and circular heads...
was proposed by H. Vogel in 1979.
This has the form
where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle
Golden angle
In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section; that is, into two arcs such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full...
, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j which depends on r, the distance from the center. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.
The bee ancestry code
Fibonacci numbers also appear in the description of the reproduction of a population of idealized honeybees, according to the following rules: If an egg is laid by an unmated female, it hatches a male or drone beeDrone (bee)Drones are male honey bees. They develop from eggs that have not been fertilized, and they cannot sting, since the worker bee's stinger is a modified ovipositor .Etymology:...
.  If, however, an egg was fertilized by a male, it hatches a female.
Thus, a male bee will always have one parent, and a female bee will have two.
If one traces the ancestry of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 greatgrandparents, 5 greatgreatgrandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, F_{n}, is the number of female ancestors, which is F_{n−1}, plus the number of male ancestors, which is F_{n−2}. (This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.)
Generalizations
The Fibonacci sequence has been generalized in many ways. These include: Generalizing the index to negative integers to produce the Negafibonacci numbers.
 Generalizing the index to real numbers using a modification of Binet's formula.
 Starting with other integers. Lucas numberLucas numberThe Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...
s have L_{1} = 1, L_{2} = 3, and L_{n} = L_{n−1} + L_{n−2}. Primefree sequencePrimefree sequenceIn mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it generally means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be...
s use the Fibonacci recursion with other starting points in order to generate sequences in which all numbers are compositeComposite numberA composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
.  Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numberPell numberIn mathematics, the Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers...
s have P_{n} = 2P_{n – 1} + P_{n – 2}.  Not adding the immediately preceding numbers. The Padovan sequencePadovan sequenceThe Padovan sequence is the sequence of integers P defined by the initial valuesP=P=P=1,and the recurrence relationP=P+P.The first few values of P are...
and Perrin numberPerrin numberIn mathematics, the Perrin numbers are defined by the recurrence relationandThe sequence of Perrin numbers starts withThe number of different maximal independent sets in an nvertex cycle graph is counted by the nth Perrin number.History:...
s have P(n) = P(n – 2) + P(n – 3).  Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as nStep Fibonacci numbers.
 Adding other objects than integers, for example functions or strings—one essential example is Fibonacci polynomials.
See also
 Collatz conjectureCollatz conjectureThe Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture , Kakutani's problem , the Thwaites conjecture , Hasse's algorithm The Collatz conjecture is a...
 Fibonacci wordFibonacci wordthumb350pxCharacterization by a [[cutting sequence]] with a line of slope 1/\varphi or \varphi1, with \varphi the [[golden ratio]].A Fibonacci word is a specific sequence of binary digits...
 HelicoidHelicoidThe helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...
 Lucas numbers
 The Fibonacci AssociationThe Fibonacci AssociationThe Fibonacci Association is a mathematical organization that specializes in the Fibonacci number sequence and a wide variety of related subjects, generalizations, and applications, including recurrence relations, combinatorial identities, binomial coefficients, prime numbers, pseudoprimes,...
 Recursion (computer science)#Fibonacci