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In mathematics, the Fibonacci numbers, commonly denoted , form a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in
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 (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
, as early as 200 BC in work by
Pingala Acharya Pingala ('; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody. The ' is a work of eight chapters in the late ...
on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
, who introduced the sequence to Western European mathematics in his 1202 book ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describe ...
''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''
Fibonacci Quarterly The ''Fibonacci Quarterly'' is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding ...
''. Applications of Fibonacci numbers include computer algorithms such as the
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. Note that the running time analysis is this ...
and the
Fibonacci heap In computer science, a Fibonacci heap is a data structure for priority queue operations, consisting of a collection of heap-ordered trees. It has a better amortized running time than many other priority queue data structures including the binar ...
data structure, and graphs called
Fibonacci cube In the mathematical field of graph theory, 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 ...
s used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a
pineapple The pineapple (''Ananas comosus'') is a tropical plant with an edible fruit; it is the most economically significant plant in the family Bromeliaceae. The pineapple is indigenous to South America, where it has been cultivated for many centuri ...
, the flowering of an
artichoke The globe artichoke (''Cynara cardunculus'' var. ''scolymus'' ),Rottenberg, A., and D. Zohary, 1996: "The wild ancestry of the cultivated artichoke." Genet. Res. Crop Evol. 43, 53–58. also known by the names French artichoke and green articho ...
, an uncurling
fern A fern (Polypodiopsida or Polypodiophyta ) is a member of a group of vascular plants (plants with xylem and phloem) that reproduce via spores and have neither seeds nor flowers. The polypodiophytes include all living pteridophytes except t ...
, and the arrangement of a
pine cone A conifer cone (in formal botanical usage: strobilus, plural strobili) is a seed-bearing organ on gymnosperm plants. It is usually woody, ovoid to globular, including scales and bracts arranged around a central axis, especially in conifers a ...
's bracts. Fibonacci numbers are also strongly related to the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
:
Binet's formula In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
expresses the th Fibonacci number in terms of and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as increases. Fibonacci numbers are also closely related to
Lucas number The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci nu ...
s, which obey the same
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
and with the Fibonacci numbers form a complementary pair of
Lucas sequence In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this recu ...
s.


Definition

The Fibonacci numbers may be defined by the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
F_0=0,\quad F_1= 1, and F_n=F_ + F_ for . Under some older definitions, the value F_0 = 0 is omitted, so that the sequence starts with F_1=F_2=1, and the recurrence F_n=F_ + F_ is valid for . The first 20 Fibonacci numbers are: :


History

The Fibonacci sequence appears in
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 (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
, in connection with
Sanskrit prosody Sanskrit prosody or Chandas refers to one of the six Vedangas, or limbs of Vedic studies.James Lochtefeld (2002), "Chandas" in The Illustrated Encyclopedia of Hinduism, Vol. 1: A-M, Rosen Publishing, , page 140 It is the study of poetic metr ...
. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration units is . Knowledge of the Fibonacci sequence was expressed as early as
Pingala Acharya Pingala ('; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody. The ' is a work of eight chapters in the late ...
( 450 BC–200 BC). Singh cites Pingala's cryptic formula ''misrau cha'' ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for beats () is obtained by adding one to the cases and one to the cases.
Bharata Muni Bharata Muni (Hindi: भरत मुनि) was an ancient sage who the musical treatise '' Natya Shastra'' is traditionally attributed to. The work covers ancient Indian dramaturgy and histrionics, especially Sanskrit theatre. Bharata is con ...
also expresses knowledge of the sequence in the ''
Natya Shastra The ''Nāṭya Śāstra'' (, ''Nāṭyaśāstra'') is a Sanskrit treatise on the performing arts. The text is attributed to sage Bharata Muni, and its first complete compilation is dated to between 200 BCE and 200 CE, but estimates vary ...
'' (c. 100 BC–c. 350 AD). However, the clearest exposition of the sequence arises in the work of
Virahanka Virahanka (Devanagari: विरहाङ्क) was an Indian prosodist who is also known for his work on mathematics. He may have lived in the 6th century, but it is also possible that he worked as late as the 8th century. His work on prosody ...
(c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):
Variations of two earlier meters s the variation.. For example, for meter of lengthfour, variations of meters of two ndthree being mixed, five happens. orks out examples 8, 13, 21.. In this way, the process should be followed in all ''mātrā-vṛttas'' rosodic combinations
Hemachandra Hemachandra was a 12th century () Indian Jain saint, scholar, poet, mathematician, philosopher, yogi, grammarian, law theorist, historian, lexicographer, rhetorician, logician, and prosodist. Noted as a prodigy by his contemporaries, he gain ...
(c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta." Outside India, the Fibonacci sequence first appears in the book ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describe ...
'' (''The Book of Calculation'', 1202) by
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
where it is used to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized (biologically unrealistic)
rabbit Rabbits, also known as bunnies or bunny rabbits, are small mammals in the family Leporidae (which also contains the hares) of the order Lagomorpha (which also contains the pikas). ''Oryctolagus cuniculus'' includes the European rabbit speci ...
population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: 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 they produce a new pair, so there are 2 pairs in the field. * At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all. * At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs. At the end of the th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month ) plus the number of pairs alive last month (month ). The number in the th month is the th Fibonacci number. The name "Fibonacci sequence" was first used by the 19th-century number theorist
Édouard Lucas __NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) 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 Lucas ...
.


Relation to the golden ratio


Closed-form expression

Like every sequence defined by a
linear recurrence with constant coefficients In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear ...
, the Fibonacci numbers have a
closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
. It has become known as Binet's formula, named after French mathematician
Jacques Philippe Marie Binet Jacques Philippe Marie Binet (; 2 February 1786 – 12 May 1856) 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 founda ...
, though it was already known by
Abraham de Moivre Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He moved ...
and
Daniel Bernoulli Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechan ...
: F_n = \frac = \frac, where \varphi = \frac \approx 1.61803\,39887\ldots is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
, and is its conjugate: \psi = \frac = 1 - \varphi = - \approx -0.61803\,39887\ldots. Since \psi = -\varphi^, this formula can also be written as F_n = \frac = \frac. To see the relation between the sequence and these constants, note that and are both solutions of the equation x^2 = x + 1 \quad\text\quad x^n = x^ + x^, so the powers of and satisfy the Fibonacci recursion. In other words, \varphi^n = \varphi^ + \varphi^ and \psi^n = \psi^ + \psi^. It follows that for any values and , the sequence defined by U_n=a \varphi^n + b \psi^n satisfies the same recurrence. U_n = a\varphi^n + b\psi^n = a(\varphi^ + \varphi^) + b(\psi^ + \psi^) = a\varphi^ + b\psi^ + a\varphi^ + b\psi^ = U_ + U_ If and are chosen so that and then the resulting sequence must be the Fibonacci sequence. This is the same as requiring and satisfy the system of equations: \left\{\begin{array}{l} a + b = 0\\ \varphi a + \psi b = 1\end{array}\right. which has solution a = \frac{1}{\varphi-\psi} = \frac{1}{\sqrt 5},\quad b = -a, producing the required formula. Taking the starting values and to be arbitrary constants, a more general solution is: U_n=a\varphi^n+b\psi^n where a=\frac{U_1-U_0\psi}{\sqrt 5} b=\frac{U_0\varphi-U_1}{\sqrt 5}.


Computation by rounding

Since \left, \frac{\psi^{n{\sqrt 5}\ < \frac{1}{2} for all , the number is the closest integer to \frac{\varphi^n}{\sqrt 5}. Therefore, it can be found by
rounding Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression with . Rounding is often done to obta ...
, using the nearest integer function: F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5}\right\rceil,\ n \geq 0. In fact, the rounding error is very small, being less than 0.1 for , and less than 0.01 for . Fibonacci numbers can also be computed by
truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
, in terms of the
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
: F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\right\rfloor,\ n \geq 0. As the floor function is
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
, the latter formula can be inverted for finding the index of the smallest Fibonacci number that is not less than a positive integer : n(F) = \left\lceil \log_\varphi \left(F\cdot\sqrt{5} - \frac{1}{2}\right) \right\rceil, where \log_\varphi(x) = \ln(x)/\ln(\varphi) = \log_{10}(x)/\log_{10}(\varphi), \ln(\varphi) = 0.481211\ldots, and \log_{10}(\varphi) = 0.208987\ldots.


Magnitude

Since ''Fn'' is
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
to \varphi^n/\sqrt5, the number of digits in ''F''''n'' is asymptotic to n\log_{10}\varphi\approx 0.2090\, n. As a consequence, for every integer ''d'' > 1 there are either 4 or 5 Fibonacci numbers with ''d'' decimal digits. More generally, in the base ''b'' representation, the number of digits in ''F''''n'' is asymptotic to n\log_b\varphi.


Limit of consecutive quotients

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
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 these ratios approach the golden ratio \varphi\colon \lim_{n\to\infty}\frac{F_{n+1{F_n}=\varphi. This convergence holds regardless of the starting values U_0 and U_1, unless U_1 = -U_0/\varphi. This can be verified using
Binet's formula In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio. In general, \lim_{n\to\infty}\frac{F_{n+m{F_n}=\varphi^m , because the ratios between consecutive Fibonacci numbers approaches \varphi. :


Decomposition of powers

Since the golden ratio satisfies the equation \varphi^2 = \varphi + 1, this expression can be used to decompose higher powers \varphi^n as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of \varphi and 1. The resulting
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
ships yield Fibonacci numbers as the linear coefficients: \varphi^n = F_n\varphi + F_{n-1}. This equation can be proved by
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
on : \varphi^{n+1} = (F_n\varphi + F_{n-1})\varphi = F_n\varphi^2 + F_{n-1}\varphi = F_n(\varphi+1) + F_{n-1}\varphi = (F_n + F_{n-1})\varphi + F_n = F_{n+1}\varphi + F_n. For \psi = -1/\varphi, it is also the case that \psi^2 = \psi + 1 and it is also the case that \psi^n = F_n\psi + F_{n-1}. These expressions are also true for if the Fibonacci sequence ''Fn'' is extended to negative integers using the Fibonacci rule F_n = F_{n+2} - F_{n+1}.


Identification

Binet's formula provides a proof that a positive integer ''x'' is a Fibonacci number if and only if at least one of 5x^2+4 or 5x^2-4 is a perfect square. This is because Binet's formula, which can be written as F_n = (\varphi^n - (-1)^n \varphi^{-n}) / \sqrt{5}, can be multiplied by \sqrt{5} \varphi^n and solved as a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
in \varphi^n via the
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, gr ...
: \varphi^n = \frac{F_n\sqrt{5} \pm \sqrt{5{F_n}^2 + 4(-1)^n{2}. Comparing this to \varphi^n = F_n \varphi + F_{n-1} = (F_n\sqrt{5} + F_n + 2 F_{n-1})/2, it follows that :5{F_n}^2 + 4(-1)^n = (F_n + 2F_{n-1})^2\,. In particular, the left-hand side is a perfect square.


Matrix form

A 2-dimensional system of linear
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s that describes the Fibonacci sequence is {F_{k+2} \choose F_{k+1 = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k alternatively denoted \vec F_{k+1} = \mathbf{A} \vec F_{k}, which yields \vec F_{n} = \mathbf{A}^n \vec F_{0}. The
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the matrix are \varphi=\frac12(1+\sqrt5) and \psi=-\varphi^{-1}=\frac12(1-\sqrt5) corresponding to the respective
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s \vec \mu={\varphi \choose 1} and \vec\nu={-\varphi^{-1} \choose 1}. As the initial value is \vec F_0={1 \choose 0}=\frac{1}{\sqrt{5\vec{\mu}-\frac{1}{\sqrt{5\vec{\nu}, it follows that the th term is \begin{align}\vec F_{n} &= \frac{1}{\sqrt{5A^n\vec\mu-\frac{1}{\sqrt{5A^n\vec\nu \\ &= \frac{1}{\sqrt{5\varphi^n\vec\mu-\frac{1}{\sqrt{5(-\varphi)^{-n}\vec\nu~\\ & =\cfrac{1}{\sqrt{5\left(\cfrac{1+\sqrt{5{2}\right)^n{\varphi \choose 1}-\cfrac{1}{\sqrt{5\left(\cfrac{1-\sqrt{5{2}\right)^n{-\varphi^{-1}\choose 1}, \end{align} From this, the th element in the Fibonacci series may be read off directly as a
closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
: F_{n} = \cfrac{1}{\sqrt{5\left(\cfrac{1+\sqrt{5{2}\right)^n-\cfrac{1}{\sqrt{5\left(\cfrac{1-\sqrt{5{2}\right)^n. Equivalently, the same computation may performed by diagonalization of through use of its
eigendecomposition In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...
: \begin{align} A & = S\Lambda S^{-1} ,\\ A^n & = S\Lambda^n S^{-1}, \end{align} where \Lambda=\begin{pmatrix} \varphi & 0 \\ 0 & -\varphi^{-1} \end{pmatrix} and S=\begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}. The closed-form expression for the th element in the Fibonacci series is therefore given by \begin{align} {F_{n+1} \choose F_{n & = A^{n} {F_1 \choose F_0} \\ & = S \Lambda^n S^{-1} {F_1 \choose F_0} \\ & = S \begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix} S^{-1} {F_1 \choose F_0} \\ & = \begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix} \begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix} \frac{1}{\sqrt{5\begin{pmatrix} 1 & \varphi^{-1} \\ -1 & \varphi \end{pmatrix} {1 \choose 0}, \end{align} which again yields F_{n} = \cfrac{\varphi^n-(-\varphi)^{-n{\sqrt{5. The matrix has a
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of −1, and thus it is a 2×2
unimodular matrix In mathematics, a unimodular matrix ''M'' is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix ''N'' that is its inverse (these are equiv ...
. This property can be understood in terms of the
continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
representation for the golden ratio: \varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}. 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-form expression for the Fibonacci numbers: \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}. For a given , this matrix can be computed in arithmetic operations, using the
exponentiation by squaring Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
method. Taking the determinant of both sides of this equation yields Cassini's identity, (-1)^n = F_{n+1}F_{n-1} - {F_n}^2. Moreover, since for any square matrix , the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing into ), \begin{align} {F_m}{F_n} + {F_{m-1{F_{n-1 &= F_{m+n-1},\\ F_{m} F_{n+1} + F_{m-1} F_n &= F_{m+n} . \end{align} In particular, with , \begin{array}{ll} F_{2 n-1} &= {F_n}^2 + {F_{n-1^2\\ F_{2 n} &= (F_{n-1}+F_{n+1})F_n\\ &= (2 F_{n-1}+F_n)F_n\\ &= (2 F_{n+1}-F_n)F_n. \end{array} These last two identities provide a way to compute Fibonacci numbers
recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
in arithmetic operations and in time , where is the time for the multiplication of two numbers of digits. This matches the time for computing the th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with
memoization In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again. Memoization ...
).


Combinatorial identities


Combinatorial proofs

Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that F_n can be interpreted as the number of ossibly emptysequences of 1s and 2s whose sum is n-1. This can be taken as the definition of F_{n} with the conventions F_{0}=0, meaning no such sequence exists whose sum is −1, and F_{1}=1, meaning the empty sequence "adds up" to 0. In the following, , {...}, is the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of a set: : F_{0} = 0 = , \{\}, : F_{1} = 1 = , \{\{\}\}, : F_{2} = 1 = , \{\{1\}\}, : F_{3} = 2 = , \{\{1,1\},\{2\}\}, : F_{4} = 3 = , \{\{1,1,1\},\{1,2\},\{2,1\}\}, : F_{5} = 5 = , \{\{1,1,1,1\},\{1,1,2\},\{1,2,1\},\{2,1,1\},\{2,2\}\}, In this manner the recurrence relation F_{n} = F_{n-1} + F_{n-2} may be understood by dividing the F_{n} sequences into two non-overlapping sets where all sequences either begin with 1 or 2: F_{n} = , \{\{1,...\},\{1,...\},...\}, + , \{\{2,...\},\{2,...\},...\}, Excluding the first element, the remaining terms in each sequence sum to n-2 or n-3 and the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of each set is F_{n-1} or F_{n-2} giving a total of F_{n-1}+F_{n-2} sequences, showing this is equal to F_{n}. In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the ''n''th is equal to the (''n'' + 2)-nd Fibonacci number minus 1. In symbols: \sum_{i=1}^n F_i = F_{n+2} - 1 This may be seen by dividing all sequences summing to n+1 based on the location of the first 2. Specifically, each set consists of those sequences that start \{2,...\}, \{1,2,...\}, ..., until the last two sets \{\{1,1,...,1,2\}\}, \{\{1,1,...,1\}\} each with cardinality 1. Following the same logic as before, by summing the cardinality of each set we see that : F_{n+2} = F_{n} + F_{n-1} + ... + , \{\{1,1,...,1,2\}\}, + , \{\{1,1,...,1\}\}, ... where the last two terms have the value F_{1} = 1. From this it follows that \sum_{i=1}^n F_i = F_{n+2}-1. A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities: \sum_{i=0}^{n-1} F_{2 i+1} = F_{2 n} and \sum_{i=1}^{n} F_{2 i} = F_{2 n+1}-1. In words, the sum of the first Fibonacci numbers with odd index up to F_{2 n-1} is the (2''n'')th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to F_{2 n} is the (2''n'' + 1)th Fibonacci number minus 1. A different trick may be used to prove \sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1} or in words, the sum of the squares of the first Fibonacci numbers up to F_{n} is the product of the ''n''th and (''n'' + 1)th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size F_{n} \times F_{n+1} and decompose it into squares of size F_{n}, F_{n-1}, ..., F_{1}; from this the identity follows by comparing areas:


Symbolic method

The sequence (F_n)_{n\in\mathbb N} is also considered using the
symbolic method In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, Siegfried Heinrich Aronhold, Alfred Clebsch, and Paul Gordan in the 19th century for computing invariant (mathematics), invariants of algebraic ...
. More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is \operatorname{Seq}(\mathcal{Z+Z^2}). Indeed, as stated above, the n-th Fibonacci number equals the number of combinatorial compositions (ordered
partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
) of n-1 using terms 1 and 2. It follows that the
ordinary generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series ...
of the Fibonacci sequence, i.e. \sum_{i=0}^\infty F_iz^i, is the complex function \frac{z}{1-z-z^2}.


Induction proofs

Fibonacci identities often can be easily proved using
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
. For example, reconsider \sum_{i=1}^n F_i = F_{n+2} - 1. Adding F_{n+1} to both sides gives :\sum_{i=1}^n F_i + F_{n+1} = F_{n+1} + F_{n+2} - 1 and so we have the formula for n+1 \sum_{i=1}^{n+1} F_i = F_{n+3} - 1 Similarly, add {F_{n+1^2 to both sides of \sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1} to give \sum_{i=1}^n {F_i}^2 + {F_{n+1^2 = F_{n+1}\left(F_{n} + F_{n+1}\right) \sum_{i=1}^{n+1} {F_i}^2 = F_{n+1}F_{n+2}


Binet formula proofs

The Binet formula is \sqrt5F_n = \varphi^n - \psi^n. This can be used to prove Fibonacci identities. For example, to prove that \sum_{i=1}^n F_i = F_{n+2} - 1 note that the left hand side multiplied by \sqrt5 becomes \begin{align} 1 +& \varphi + \varphi^2 + \dots + \varphi^n - \left(1 + \psi + \psi^2 + \dots + \psi^n \right)\\ &= \frac{\varphi^{n+1}-1}{\varphi-1} - \frac{\psi^{n+1}-1}{\psi-1}\\ &= \frac{\varphi^{n+1}-1}{-\psi} - \frac{\psi^{n+1}-1}{-\varphi}\\ &= \frac{-\varphi^{n+2}+\varphi + \psi^{n+2}-\psi}{\varphi\psi}\\ &= \varphi^{n+2}-\psi^{n+2}-(\varphi-\psi)\\ &= \sqrt5(F_{n+2}-1)\\ \end{align} as required, using the facts \varphi\psi =- 1 and \varphi-\psi=\sqrt5 to simplify the equations.


Other identities

Numerous other identities can be derived using various methods. Here are some of them:


Cassini's and Catalan's identities

Cassini's identity states that {F_n}^2 - F_{n+1}F_{n-1} = (-1)^{n-1} Catalan's identity is a generalization: {F_n}^2 - F_{n+r}F_{n-r} = (-1)^{n-r}{F_r}^2


d'Ocagne's identity

F_m F_{n+1} - F_{m+1} F_n = (-1)^n F_{m-n} F_{2 n} = {F_{n+1^2 - {F_{n-1^2 = F_n \left (F_{n+1}+F_{n-1} \right ) = F_nL_n where ''L''''n'' is the ''nth
Lucas number The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci nu ...
. The last is an identity for doubling ''n''; other identities of this type are F_{3 n} = 2{F_n}^3 + 3 F_n F_{n+1} F_{n-1} = 5{F_n}^3 + 3 (-1)^n F_n by Cassini's identity. F_{3 n+1} = {F_{n+1^3 + 3 F_{n+1}{F_n}^2 - {F_n}^3 F_{3 n+2} = {F_{n+1^3 + 3 {F_{n+1^2 F_n + {F_n}^3 F_{4 n} = 4 F_n F_{n+1} \left ({F_{n+1^2 + 2{F_n}^2 \right ) - 3{F_n}^2 \left ({F_n}^2 + 2{F_{n+1^2 \right ) These can be found experimentally using
lattice reduction In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different algorithms, whose running time is usually at least exponent ...
, and are useful in setting up the
special number field sieve In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for intege ...
to
factorize In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind ...
a Fibonacci number. More generally, F_{k n+c} = \sum_{i=0}^k {k\choose i} F_{c-i} {F_n}^i {F_{n+1^{k-i}. or alternatively F_{k n+c} = \sum_{i=0}^k {k\choose i} F_{c+i} {F_n}^i {F_{n-1^{k-i}. Putting in this formula, one gets again the formulas of the end of above section Matrix form.


Generating function

The
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
of the Fibonacci sequence is the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
s(x)=\sum_{k=0}^{\infty} F_k x^k = \sum_{k=1}^{\infty} F_k x^k = 0+x+x^2+2 x^3+3 x^4+\dots. This series is convergent for , x, < \frac{1}{\varphi}, and its sum has a simple closed-form: s(x)=\frac{x}{1-x-x^2} This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum: \begin{align} s(x) &= \sum_{k=0}^{\infty} F_k x^k \\ &= F_0 + F_1x + \sum_{k=2}^{\infty} F_k x^k \\ &= 0 + 1x + \sum_{k=2}^{\infty} F_k x^k \\ &= x + \sum_{k=2}^{\infty} \left( F_{k-1} + F_{k-2} \right) x^k \\ &= x + \sum_{k=2}^{\infty} F_{k-1} x^k + \sum_{k=2}^{\infty} F_{k-2} x^k \\ &= x + x\sum_{k=2}^{\infty} F_{k-1} x^{k-1} + x^2\sum_{k=2}^{\infty} F_{k-2} x^{k-2} \\ &= x + x\sum_{k=1}^{\infty} F_k x^k + x^2\sum_{k=0}^{\infty} F_k x^k \\ &= x + x s(x) + x^2 s(x). \end{align} Solving the equation s(x)=x+xs(x)+x^2 s(x) for ''s''(''x'') results in the closed form. The
partial fraction decomposition In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
is given by s(x) = \frac{1}{\sqrt5}\left(\frac{1}{1 - \varphi x} - \frac{1}{1 - \psi x}\right) where \varphi = \frac{1 + \sqrt{5{2} is the golden ratio and \psi = \frac{1 - \sqrt{5{2} is its conjugate. -s\left(-\frac{1}{x}\right) gives the generating function for the negafibonacci numbers, and s(x) satisfies the
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
s(x)=s\left(-\frac{1}{x}\right).


Reciprocal sums

Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of
theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
s. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as \sum_{k=1}^\infty \frac{1}{F_{2 k-1 = \frac{\sqrt{5{4} \;\, \vartheta_2\!\left(0, \frac{3-\sqrt 5}{2}\right)^2 , and the sum of squared reciprocal Fibonacci numbers as \sum_{k=1}^\infty \frac{1} = \frac{\sqrt{5{2}, and there is a ''nested'' sum of squared Fibonacci numbers giving the reciprocal of the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
, \sum_{k=1}^\infty \frac{(-1)^{k+1{\sum_{j=1}^k {F_{j^2} = \frac{\sqrt{5}-1}{2} . The sum of all even-indexed reciprocal Fibonacci numbers is \sum_{k=1}^{\infty} \frac{1}{F_{2 k = \sqrt{5} \left(L\bigl(\psi^2\bigr) - L\bigl(\psi^4\bigr)\right) with the
Lambert series In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form :S(q)=\sum_^\infty a_n \frac . It can be resumed formally by expanding the denominator: :S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m ...
  \textstyle L(q) := \sum_{k=1}^{\infty} \frac{q^k}{1-q^k} ,   since   \textstyle \frac{1}{F_{2 k = \sqrt{5} \left(\frac{\psi^{2 k{1-\psi^{2 k - \frac{\psi^{4 k{1-\psi^{4 k \right). So the
reciprocal Fibonacci constant The reciprocal Fibonacci constant, or ψ, is defined as the sum of the reciprocals of the Fibonacci numbers: :\psi = \sum_^ \frac = \frac + \frac + \frac + \frac + \frac + \frac + \frac + \frac + \cdots. The ratio of successive terms in this su ...
is \sum_{k=1}^{\infty} \frac{1}{F_k} = \sum_{k=1}^\infty \frac{1}{F_{2 k-1 + \sum_{k=1}^{\infty} \frac {1}{F_{2 k = 3.359885666243 \dots Moreover, this number has been proved
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
by
Richard André-Jeannin Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'stron ...
. Millin's series gives the identity \sum_{k=0}^{\infty} \frac{1}{F_{2^k = \frac{7 - \sqrt{5{2}, which follows from the closed form for its partial sums as ''N'' tends to infinity: \sum_{k=0}^N \frac{1}{F_{2^k = 3 - \frac{F_{2^N-1{F_{2^N.


Primes and divisibility


Divisibility properties

Every third number of the sequence is even (a multiple of F_3=2) and, more generally, every ''k''th number of the sequence is a multiple of ''Fk''. Thus the Fibonacci sequence is an example of a
divisibility sequence In mathematics, a divisibility sequence is an integer sequence (a_n) indexed by positive integers ''n'' such that :\textm\mid n\texta_m\mid a_n for all ''m'', ''n''. That is, whenever one index is a multiple of another one, then the co ...
. In fact, the Fibonacci sequence satisfies the stronger divisibility property \gcd(F_a,F_b,F_c,\ldots) = F_{\gcd(a,b,c,\ldots)}\, where is the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
function. In particular, any three consecutive Fibonacci numbers are pairwise
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
because both F_1=1 and F_2 = 1. That is, :\gcd(F_n, F_{n+1}) = \gcd(F_n, F_{n+2}) = \gcd(F_{n+1}, F_{n+2}) = 1 for every ''n''. Every prime number ''p'' divides a Fibonacci number that can be determined by the value of ''p'' modulo 5. If ''p'' is congruent to 1 or 4 (mod 5), then ''p'' divides ''F''''p'' − 1, and if ''p'' is congruent to 2 or 3 (mod 5), then, ''p'' divides ''F''''p'' + 1. The remaining case is that ''p'' = 5, and in this case ''p'' divides ''F''p. \begin{cases} p =5 & \Rightarrow p \mid F_{p}, \\ p \equiv \pm1 \pmod 5 & \Rightarrow p \mid F_{p-1}, \\ p \equiv \pm2 \pmod 5 & \Rightarrow p \mid F_{p+1}.\end{cases} These cases can be combined into a single, non-
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
formula, using the
Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
: p \mid F_{p-\left(\frac{5}{p}\right)}.


Primality testing

The above formula can be used as a primality test in the sense that if n \mid F_{n-\left(\frac{5}{n}\right)}, where the Legendre symbol has been replaced by the
Jacobi symbol Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a ...
, then this is evidence that ''n'' is a prime, and if it fails to hold, then ''n'' is definitely not a prime. If ''n'' is composite and satisfies the formula, then ''n'' is a ''Fibonacci pseudoprime''. When ''m'' is largesay a 500-bit numberthen we can calculate ''F''''m'' (mod ''n'') efficiently using the matrix form. Thus \begin{pmatrix} F_{m+1} & F_m \\ F_m & F_{m-1} \end{pmatrix} \equiv \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^m \pmod n. Here the matrix power ''A''''m'' is calculated using
modular exponentiation Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys. Modular ...
, which can be adapted to matrices.


Fibonacci primes

A ''Fibonacci prime'' is a Fibonacci number that is
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. 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 runs of
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s, there are therefore also arbitrarily long runs of composite Fibonacci numbers. No Fibonacci number greater than ''F''6 = 8 is one greater or one less than a prime number. The only nontrivial
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
Fibonacci number is 144. Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers. 1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming. No Fibonacci number can be a
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
. More generally, no Fibonacci number other than 1 can be multiply perfect, and no ratio of two Fibonacci numbers can be perfect.


Prime divisors

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 In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind ''U'n''(''P'', ''Q'') with relatively prime parameters ''P'',  ...
). As a result, 8 and 144 (''F''6 and ''F''12) are the only Fibonacci numbers that are the product of other Fibonacci numbers. The divisibility of Fibonacci numbers by a prime ''p'' is related to the
Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
\left(\tfrac{p}{5}\right) which is evaluated as follows: \left(\frac{p}{5}\right) = \begin{cases} 0 & \text{if } p = 5\\ 1 & \text{if } p \equiv \pm 1 \pmod 5\\ -1 & \text{if } p \equiv \pm 2 \pmod 5.\end{cases} If ''p'' is a prime number then F_p \equiv \left(\frac{p}{5}\right) \pmod p \quad \text{and}\quad F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p. For example, \begin{align} (\tfrac{2}{5}) &= -1, &F_3 &= 2, &F_2&=1, \\ (\tfrac{3}{5}) &= -1, &F_4 &= 3,&F_3&=2, \\ (\tfrac{5}{5}) &= 0, &F_5 &= 5, \\ (\tfrac{7}{5}) &= -1, &F_8 &= 21,&F_7&=13, \\ (\tfrac{11}{5})& = +1, &F_{10}& = 55, &F_{11}&=89. \end{align} It is not known whether there exists a prime ''p'' such that F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod{p^2}. Such primes (if there are any) would be called
Wall–Sun–Sun prime In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. Definition Let p be a prime number. When each term in the sequence of Fibonac ...
s. Also, if ''p'' ≠ 5 is an odd prime number then: 5 {F_{\frac{p \pm 1}{2}^2 \equiv \begin{cases} \tfrac{1}{2} \left (5\left(\frac{p}{5}\right)\pm 5 \right ) \pmod p & \text{if } p \equiv 1 \pmod 4\\ \tfrac{1}{2} \left (5\left(\frac{p}{5}\right)\mp 3 \right ) \pmod p & \text{if } p \equiv 3 \pmod 4. \end{cases} Example 1. ''p'' = 7, in this case ''p'' ≡ 3 (mod 4) and we have: (\tfrac{7}{5}) = -1: \qquad \tfrac{1}{2}\left (5(\tfrac{7}{5})+3 \right ) =-1, \quad \tfrac{1}{2} \left (5(\tfrac{7}{5})-3 \right )=-4. F_3=2 \text{ and } F_4=3. 5{F_3}^2=20\equiv -1 \pmod {7}\;\;\text{ and }\;\;5{F_4}^2=45\equiv -4 \pmod {7} Example 2. ''p'' = 11, in this case ''p'' ≡ 3 (mod 4) and we have: (\tfrac{11}{5}) = +1: \qquad \tfrac{1}{2}\left (5(\tfrac{11}{5})+3 \right )=4, \quad \tfrac{1}{2} \left (5(\tfrac{11}{5})- 3 \right )=1. F_5=5 \text{ and } F_6=8. 5{F_5}^2=125\equiv 4 \pmod {11} \;\;\text{ and }\;\;5{F_6}^2=320\equiv 1 \pmod {11} Example 3. ''p'' = 13, in this case ''p'' ≡ 1 (mod 4) and we have: (\tfrac{13}{5}) = -1: \qquad \tfrac{1}{2}\left (5(\tfrac{13}{5})-5 \right ) =-5, \quad \tfrac{1}{2}\left (5(\tfrac{13}{5})+ 5 \right )=0. F_6=8 \text{ and } F_7=13. 5{F_6}^2=320\equiv -5 \pmod {13} \;\;\text{ and }\;\;5{F_7}^2=845\equiv 0 \pmod {13} Example 4. ''p'' = 29, in this case ''p'' ≡ 1 (mod 4) and we have: (\tfrac{29}{5}) = +1: \qquad \tfrac{1}{2}\left (5(\tfrac{29}{5})-5 \right )=0, \quad \tfrac{1}{2}\left (5(\tfrac{29}{5})+5 \right )=5. F_{14}=377 \text{ and } F_{15}=610. 5{F_{14^2=710645\equiv 0 \pmod {29} \;\;\text{ and }\;\;5{F_{15^2=1860500\equiv 5 \pmod {29} For odd ''n'', all odd prime divisors of ''F''''n'' are congruent to 1 modulo 4, implying that all odd divisors of ''F''''n'' (as the products of odd prime divisors) are congruent to 1 modulo 4. For example, F_1 = 1, F_3 = 2, F_5 = 5, F_7 = 13, F_9 = 34 = 2 \cdot 17, F_{11} = 89, F_{13} = 233, F_{15} = 610 = 2 \cdot 5 \cdot 61. All known factors of Fibonacci numbers ''F''(''i'') for all ''i'' < 50000 are collected at the relevant repositories.


Periodicity modulo ''n''

If the members of the Fibonacci sequence are taken mod ''n'', the resulting sequence is periodic with period at most ''6n''. The lengths of the periods for various ''n'' form the so-called
Pisano period In number theory, the ''n''th Pisano period, written as '(''n''), is the period with which the sequence of Fibonacci numbers taken modulo ''n'' repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of ...
s. Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the
multiplicative order In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n. In other words, the multiplicative order ...
of a modular integer or of an element in a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
. However, for any particular ''n'', the Pisano period may be found as an instance of
cycle detection In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function that maps a finite set to itself, and any initial value in , the sequence of itera ...
.


Generalizations

The Fibonacci sequence is one of the simplest and earliest known sequences defined by a
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, and specifically by a
linear difference equation Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients. Some specific examples that are close, in some sense, from Fibonacci sequence 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 number The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci nu ...
s have ''L''1 = 1, ''L''2 = 3, and ''Ln'' = ''L''''n''−1 + ''L''''n''−2.
Primefree sequence In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initia ...
s use the Fibonacci recursion with other starting points to generate sequences in which all numbers are
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
. * Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The
Pell number In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , an ...
s have ''Pn'' = 2''P''''n'' − 1 + ''P''''n'' − 2. If the coefficient of the preceding value is assigned a variable value ''x'', the result is the sequence of
Fibonacci polynomials In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. Definition The ...
. * Not adding the immediately preceding numbers. The
Padovan sequence In number theory, the Padovan sequence is the sequence of integers ''P''(''n'') defined. by the initial values :P(0)=P(1)=P(2)=1, and the recurrence relation :P(n)=P(n-2)+P(n-3). The first few values of ''P''(''n'') are :1, 1, 1, 2, 2, 3, 4, 5 ...
and
Perrin number In mathematics, the Perrin numbers are defined by the recurrence relation : for , with initial values :. The sequence of Perrin numbers starts with : 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, ... The number of different maxima ...
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 ''n-Step Fibonacci numbers''.


Applications


Mathematics

The Fibonacci numbers occur in the sums of "shallow" diagonals in
Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
(see
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
): The generating function can be expanded into \frac{x}{1-x-x^2} = x + x^2(1+x) + x^3(1+x)^2 + \dots + x^{k+1}(1+x)^k + \dots = \sum\limits_{n=0}^\infty F_n x^n and collecting like terms of x^n, we have the identity F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}. To see how the formula is used, we can arrange the sums by the number of terms present: :{, , , , - , , , , , , - , , , , which is \binom{5}{0}+\binom{4}{1}+\binom{3}{2}, where we are choosing the positions of ''k'' twos from ''n-k-1'' terms. These numbers also give the solution to certain enumerative problems, the most common of which is that of counting the number of ways of writing a given number as an ordered sum of 1s and 2s (called
compositions Composition or Compositions may refer to: Arts and literature * Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
); there are ways to do this (equivalently, it's also the number of
domino tiling In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by p ...
s of the 2\times n rectangle). For example, there are ways one can climb a staircase of 5 steps, taking one or two steps at a time: :{, , , , , , , , - , , , , The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied
recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
until a single step, of which there is only one way to climb. The Fibonacci numbers can be found in different ways among the set of
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...
, or equivalently, among the
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of a given set. * The number of binary strings of length without consecutive s is the Fibonacci number . For example, out of the 16 binary strings of length 4, there are without consecutive s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of
Fibbinary number In mathematics, the fibbinary numbers are the numbers whose binary representation does not contain two consecutive ones. That is, they are sums of distinct and non-consecutive powers of two. Relation to binary and Fibonacci numbers The fibbinary n ...
s. Equivalently, is the number of subsets of without consecutive integers, that is, those for which for every . A
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
with the sums to ''n+1'' is to replace ''1'' with ''0'' and ''2'' with ''10'', and drop the last zero. * The number of binary strings of length without an odd number of consecutive s is the Fibonacci number . For example, out of the 16 binary strings of length 4, there are without an odd number of consecutive s – they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets of without an odd number of consecutive integers is . A bijection with the sums to ''n'' is to replace ''1'' with ''0'' and ''2'' with ''11''. * The number of binary strings of length without an even number of consecutive s or s is . For example, out of the 16 binary strings of length 4, there are without an even number of consecutive s or s – they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets. *
Yuri Matiyasevich Yuri Vladimirovich Matiyasevich, (russian: Ю́рий Влади́мирович Матиясе́вич; born 2 March 1947 in Leningrad) is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's t ...
was able to show that the Fibonacci numbers can be defined by a
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
, which led to his solving
Hilbert's tenth problem Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equat ...
. *The Fibonacci numbers are also an example of a
complete sequence In mathematics, a sequence of natural numbers 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. For example, the sequence of powers of two (1, 2, 4, 8, ...) ...
. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most. *Moreover, 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 In mathematics, Zeckendorf's theorem, named after Belgian amateur mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers. Zeckendorf's theorem states that every positive integer can be rep ...
, 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 In mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no ...
. *Starting with 5, every second Fibonacci number is the length of the hypotenuse of a
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
with integer sides, or in other words, the largest number in a
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
, obtained from the formula (F_n F_{n+3})^2 + (2 F_{n+1}F_{n+2})^2 = {F_{2 n+3^2. The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... The middle side of each of these triangles is the sum of the three sides of the preceding triangle. *The
Fibonacci cube In the mathematical field of graph theory, 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 ...
is an
undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...
with a Fibonacci number of nodes that has been proposed as a
network topology Network topology is the arrangement of the elements ( links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and contro ...
for
parallel computing Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different fo ...
. *Fibonacci numbers appear in the ring lemma, used to prove connections between the
circle packing theorem The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in gen ...
and
conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
s.


Computer science

*The Fibonacci numbers are important in computational run-time analysis of
Euclid's algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an eff ...
to determine the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers. *Fibonacci numbers are used in a polyphase version of the
merge sort In computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the order of equal elements is the same i ...
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 tape-drive implementation of the
polyphase merge sort A polyphase merge sort is a variation of a bottom-up merge sort that sorts a list using an initial uneven distribution of sub-lists (runs), primarily used for external sorting, and is more efficient than an ordinary merge sort when there are fewer ...
was described in ''
The Art of Computer Programming ''The Art of Computer Programming'' (''TAOCP'') is a comprehensive monograph written by the computer scientist Donald Knuth presenting programming algorithms and their analysis. Volumes 1–5 are intended to represent the central core of compu ...
''. *A Fibonacci tree is a
binary tree In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
whose child trees (recursively) differ in
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
by exactly 1. So it is an
AVL tree In computer science, an AVL tree (named after inventors Adelson-Velsky and Landis) is a self-balancing binary search tree. It was the first such data structure to be invented. In an AVL tree, the heights of the two child subtrees of any nod ...
, and one with the fewest nodes for a given height — the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees. *Fibonacci numbers are used by some
pseudorandom number generators A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...
. *Fibonacci numbers arise in the analysis of the
Fibonacci heap In computer science, a Fibonacci heap is a data structure for priority queue operations, consisting of a collection of heap-ordered trees. It has a better amortized running time than many other priority queue data structures including the binar ...
data structure. *A one-dimensional optimization method, called the
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. Note that the running time analysis is this ...
, uses Fibonacci numbers. *The Fibonacci number series is used for optional
lossy compression In information technology, lossy compression or irreversible compression is the class of data compression methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to reduce data size ...
in the
IFF In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ...
8SVX 8-Bit Sampled Voice (8SVX) is an audio file format standard developed by Electronic Arts for the Commodore-Amiga computer series. It is a data subtype of the IFF file container format. It typically contains linear pulse-code modulation (LPCM) ...
audio file format used on
Amiga Amiga is a family of personal computers introduced by Commodore in 1985. The original model is one of a number of mid-1980s computers with 16- or 32-bit processors, 256 KB or more of RAM, mouse-based GUIs, and significantly improved graphi ...
computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law. *Some Agile teams use a modified series called the "Modified Fibonacci Series" in
planning poker Planning poker, also called Scrum poker, is a consensus-based, gamified technique for estimating, mostly used for timeboxing in '' Agile principles''. In planning poker, members of the group make estimates by playing numbered cards face-down t ...
, as an estimation tool. Planning Poker is a formal part of the Scaled Agile Framework. *
Fibonacci coding In mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no ...
* NegaFibonacci coding


Nature

Fibonacci sequences appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a
pineapple The pineapple (''Ananas comosus'') is a tropical plant with an edible fruit; it is the most economically significant plant in the family Bromeliaceae. The pineapple is indigenous to South America, where it has been cultivated for many centuri ...
, the flowering of
artichoke The globe artichoke (''Cynara cardunculus'' var. ''scolymus'' ),Rottenberg, A., and D. Zohary, 1996: "The wild ancestry of the cultivated artichoke." Genet. Res. Crop Evol. 43, 53–58. also known by the names French artichoke and green articho ...
, an uncurling fern and the arrangement of a
pine cone A conifer cone (in formal botanical usage: strobilus, plural strobili) is a seed-bearing organ on gymnosperm plants. It is usually woody, ovoid to globular, including scales and bracts arranged around a central axis, especially in conifers a ...
, and the family tree of honeybees.
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
pointed out the presence of the Fibonacci sequence in nature, using it to explain the (
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
-related) pentagonal form of some flowers. Field daisies most often have petals in counts of Fibonacci numbers. In 1830, K. F. Schimper and A. Braun discovered that the parastichies (spiral
phyllotaxis In botany, phyllotaxis () or phyllotaxy is the arrangement of leaf, leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature. Leaf arrangement The basic leaf#Arrangement on the stem, arrangements of leaves ...
) of plants were frequently expressed as fractions involving Fibonacci numbers.
Przemysław Prusinkiewicz Przemysław (Przemek) Prusinkiewicz is a Polish computer scientist who advanced the idea that Fibonacci numbers in nature can be in part understood as the expression of certain algebraic constraints on free groups, specifically as certain Linden ...
advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
s, specifically as certain Lindenmayer grammars. A model for the pattern of
floret This glossary of botanical terms is a list of definitions of terms and concepts relevant to botany and plants in general. Terms of plant morphology are included here as well as at the more specific Glossary of plant morphology and Glossary o ...
s in the head of a
sunflower The common sunflower (''Helianthus annuus'') is a large annual forb of the genus ''Helianthus'' grown as a crop for its edible oily seeds. Apart from cooking oil production, it is also used as livestock forage (as a meal or a silage plant), as ...
was proposed by in 1979. This has the form \theta = \frac{2\pi}{\varphi^2} n,\ r = c \sqrt{n} where is the index number of the floret and is a constant scaling factor; the florets thus lie on
Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance f ...
. The divergence angle, approximately 137.51°, is the
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 ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the l ...
, 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 , the nearest neighbors of floret number are those at for some index , which depends on , the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers, typically counted by the outermost range of radii. Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules: * If an egg is laid by an unmated female, it hatches a male or drone bee. * If, however, an egg was fertilized by a male, it hatches a female. Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, , is the number of female ancestors, which is , plus the number of male ancestors, which is . This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated. It has been noticed that the number of possible ancestors on the human
X chromosome The X chromosome is one of the two sex-determining chromosomes (allosomes) in many organisms, including mammals (the other is the Y chromosome), and is found in both males and females. It is a part of the XY sex-determination system and XO sex-d ...
inheritance line at a given ancestral generation also follows the Fibonacci sequence. A male individual has an X chromosome, which he received from his mother, and a
Y chromosome The Y chromosome is one of two sex chromosomes (allosomes) in therian mammals, including humans, and many other animals. The other is the X chromosome. Y is normally the sex-determining chromosome in many species, since it is the presence or abse ...
, which he received from his father. The male counts as the "origin" of his own X chromosome (F_1=1), and at his parents' generation, his X chromosome came from a single parent (F_2=1). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome (F_3=2). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome (F_4=3). Five great-great-grandparents contributed to the male descendant's X chromosome (F_5=5), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a population founder appears on all lines of the genealogy.)


Other

*In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, when a beam of light shines at an angle through two stacked transparent plates of different materials of different
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
es, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have reflections, for , is the kth Fibonacci number. (However, when , there are three reflection paths, not two, one for each of the three surfaces.) *
Fibonacci retracement In finance, Fibonacci retracement is a method of technical analysis for determining support and resistance levels. It is named after the Fibonacci sequence of numbers, whose ratios provide price levels to which markets tend to retrace a portion ...
levels are widely used in
technical analysis In finance, technical analysis is an analysis methodology for analysing and forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and quantitative analysis use many of the sam ...
for financial market trading. *Since the
conversion Conversion or convert may refer to: Arts, entertainment, and media * "Conversion" (''Doctor Who'' audio), an episode of the audio drama ''Cyberman'' * "Conversion" (''Stargate Atlantis''), an episode of the television series * "The Conversion" ...
factor 1.609344 for miles to kilometers is close to the golden ratio, 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 In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
2 number
register Register or registration may refer to: Arts entertainment, and media Music * Register (music), the relative "height" or range of a note, melody, part, instrument, etc. * ''Register'', a 2017 album by Travis Miller * Registration (organ), th ...
in
golden ratio base Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number  ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, ...
''φ'' being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead. *The measured values of voltages and currents in the infinite resistor chain circuit (also called the
resistor ladder A resistor ladder is an electrical circuit made from repeating units of resistors. Two configurations are discussed below, a string resistor ladder and an R-2R ladder. An R–2R ladder is a simple and inexpensive way to perform digital-to-analog ...
or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio. *Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics. In particular, it is shown how a generalised Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model. *
Mario Merz Mario Merz (1 January 1925 – 9 November 2003) was an Italian artist, and husband of Marisa Merz. Life Born in Milan, Merz started drawing during World War II, when he was imprisoned for his activities with the ''Giustizia e Libertà'' antifa ...
included the Fibonacci sequence in some of his artworks beginning in 1970. *
Joseph Schillinger Joseph Moiseyevich Schillinger (Russian: Иосиф Моисеевич Шиллингер, (other sources: ) – 23 March 1943) was a composer, music theorist, and composition teacher who originated the Schillinger System of Musical Composition ...
(1895–1943) developed a system of composition which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature. See also .


See also

* * * * *


References

Footnotes Citations


Works cited

* . * . * . * * . * * . *


External links

*
Periods of Fibonacci Sequences Mod m
at MathPages

* * * {{Interwiki extra, qid=Q23835349 Articles containing proofs