In
mathematicsMathematics 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 sequenceIn 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 relationIn 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 AbaciLiber 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 mathematicsIndian 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 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 in that they are a complementary pair of
Lucas sequenceIn 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 ratioIn 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 techniqueIn 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 heapIn 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 cubeThe 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 graph-theoretic 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 stemIn botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem .- Pattern structure :...
, the fruit spouts of a
pineapplePineapple 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-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
fernA 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 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 prosodyVersification 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
PingalaPingala 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
VirahankaVirahanka 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 Chhanda-sutras 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
VirahankaVirahanka 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 Chhanda-sutras 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 mAtrA-vr.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 AbaciLiber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci...
(1202) by Leonardo of Pisa, known as
FibonacciLeonardo 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)
rabbitRabbits 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 19th-century number theorist
Édouard LucasFrançois 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:
| F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
F11 |
F12 |
F13 |
F14 |
F15 |
F16 |
F17 |
F18 |
F19 |
F20 |
| 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 re-arranged 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 |
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
| −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 triangleIn 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 coefficientIn 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
binaryThe binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
stringsIn 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 Fn+2. For example, out of the 16 binary strings of length 4, there are F6 = 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 Fn+2.
- The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number Fn+1. For example, out of the 16 binary strings of length 4, there are F5 = 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 2Fn. For example, out of the 16 binary strings of length 4, there are 2F4 = 6 without an even number of consecutive 0s or 1s – they are 0001, 1000, 1110, 0111, 0101, 1010.
Closed-form expression
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form solution. It has become known as
BinetJacques 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 MoivreAbraham 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 ratioIn 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
Since
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 functionIn 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 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 equation
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 relationIn 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 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
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
determinantIn 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 matrixIn 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 fractionIn 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, brute-force test is the identity
which is true
if and only ifIn 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 closed-form 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 squareIn 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
convergentA 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 fractionIn 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
coprimeIn 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 argumentsIn 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 non-overlapping 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:
- For
 ,  , so the equation is true for  .
- For
 , assume  .
- Add the next Fibonacci number
 to both sides:  .
- By the Fibonacci recurrence relation,
 , so  , which is the  case, proving that where the equation is true for  , so is it for  . |
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 F2n:
  
A basis case for this could be F1 = F2.
2: j is even
By induction for F2n+1:
  
A basis case for this could be F0 = F1 − 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
- [F(n + 2) − 1] + [F(n + 1) − 1] + ... + [F(2) − 1]
- = [F(n + 2) − 1] + [F(n + 1) − 1] + ... + [F(2) − 1] + [F(1) − 1] + F(0)
- = F(n + 2) + [F(n + 1) + ... + F(1) + F(0)] − (n + 2)
- = F(n + 2) + [F(n + 3) − 1] − (n + 2)
- = F(n + 2) + F(n + 3) − (n + 3).
On the other hand, we observe from the second identity that there are
- F(0) + F(1) + ... + F(n − 1) + F(n) ways summing to n + 1;
- F(0) + F(1) + ... + F(n − 1) ways summing to n;
......
Adding up all n + 1 sums, we see that there are
- (n + 1) F(0) + n F(1) + ... + F(n) ways summing to −1, 0, ..., or n + 1.
Since the two methods of counting refer to the same number, we have
- (n + 1) F(0) + n F(1) + ... + F(n) = F(n + 2) + F(n + 3) − (n + 3)
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 n-th 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 n-th 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 is
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.
Other identities
Other identities include relationships to the
Lucas 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, which have the same recursive properties but start with L
0 = 2 and L
1 = 1. These properties include F
2n = F
nL
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 reductionIn 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 sieveIn number theory, a branch of mathematics, the special number field sieve is a special-purpose integer factorization algorithm. The general number field sieve was derived from it....
to
factorizeIn 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 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 n-vertex cycle graph is counted by the nth Perrin number.-History:...
s for details.
Power series
The
generating functionIn 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 closed-form 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 puzzle-books 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 odd-indexed reciprocal Fibonacci number as
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 ratioIn 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
irrationalIn 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 non-zero, and is therefore not a rational number....
by Richard André-Jeannin.
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 property
Fibonacci primes
A Fibonacci prime is a Fibonacci number that is
primeA 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 longIn 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 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....
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 theoremCarmichael'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
squareIn 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 non-trivial 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,
- gcd
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero 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...
(Fn, Fn+1) = gcd(Fn, Fn+2) = 1.
More generally,
- gcd(Fn, Fm) = Fgcd(n, m).
Prime divisors of Fibonacci numbers
The divisibility of Fibonacci numbers by a prime p is related to the
Legendre symbolIn 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 non-residue 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,
F1 = 1, F3 = 2, F5 = 5, F7 = 13, F9 = 34 = 2×17, F11 = 89, F13 = 233, F15 = 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
periodicIn 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 so-called
Pisano periodIn 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 tripleA Pythagorean triple consists of three positive integers a, b, and c, 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 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
asymptoticIn 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 run-time analysis of
Euclid's algorithmIn 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 divisorIn mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero 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 MatiyasevichYuri 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 1962-1963 studied at Saint Petersburg Lyceum 239...
was able to show that the Fibonacci numbers can be defined by a
Diophantine equationIn 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 problemHilbert'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 sequenceIn 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 theoremZeckendorf'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 codingIn 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 sortMerge sort is an O comparison-based 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 tape-drive implementation of the
polyphase merge sortA 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 ProgrammingThe 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 heapIn 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 cubeThe 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 graph-theoretic mathematics...
is an undirected graph with a Fibonacci number of nodes that has been proposed as a
network topologyNetwork topology is the layout pattern of interconnections of the various elements of a computer or biological network....
for
parallel computingParallel 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: bit-level,...
.
A one-dimensional optimization method, called the
Fibonacci search techniqueIn 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
IFFInterchange 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....
8SVX8SVX is a subformat of the Interchange File Format. The subformat is for 8-bit 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
AmigaThe 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 high-end home computer and became popular for its graphical, audio and multi-tasking abilities...
computers. The number series
compandsIn 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 µ-law.
In
musicMusic 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 BartókBéla 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
conversionConversion 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
radixIn 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
registerIn 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 baseGolden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, 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 stemIn botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem .- Pattern structure :...
, the fruitlets of a
pineapplePineapple 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-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 groupIn 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 grammarAn L-system 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
sunflowerSunflower 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 angleIn 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 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 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, 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 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 have L1 = 1, L2 = 3, and Ln = Ln−1 + Ln−2. Primefree 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 compositeA 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 number
In 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 Pn = 2Pn – 1 + Pn – 2.
- Not adding the immediately preceding numbers. The Padovan sequence
The 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 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 n-vertex 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 n-Step Fibonacci numbers.
- Adding other objects than integers, for example functions or strings—one essential example is Fibonacci polynomials.
See also
- Collatz conjecture
The 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 word
thumb|350px|Characterization by a [[cutting sequence]] with a line of slope 1/\varphi or \varphi-1, with \varphi the [[golden ratio]].A Fibonacci word is a specific sequence of binary digits...
- Helicoid
The 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 Association
The 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
External links