In mathematics, the Fibonacci numbers, commonly denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,
Mathematics includes the study of such topics as quantity, structure, space, and change.
In mathematics, an integer sequence is a sequence of integers.
for n > 1.
One has F2 = 1. In some books, and particularly in old ones, F0, the "0" is omitted, and the Fibonacci sequence starts with F1 = F2 = 1. The beginning of the sequence is thus:
Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.
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. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,
Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. They appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci , Fibonacci introduced the sequence to Western European mathematics,although the sequence had been described earlier in Indian mathematics.
Fibonacci was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, "Fibonacci", was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for filius Bonacci. He is also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo ("traveller") Pisano.
Acharya Pingala was an ancient Indian mathematician who authored the Chandaḥśāstra, the earliest known treatise on Sanskrit prosody.
Liber Abaci is a 1202 historic book on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci.
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly . Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes 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 flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.
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 editors were Verner Emil Hoggatt, Jr. and Alfred Brousseau; the present editor is Professor Curtis Cooper of the Mathematics Department of the University of Central Missouri.
In computer science, the Fibonacci search technique is a method of searching a sorted array using a divide and conquer algorithm that narrows down possible locations with the aid of Fibonacci numbers. Compared to binary search where the sorted array is divided into two equal-sized parts, one of which is examined further, Fibonacci search divides the array into two parts that have sizes that are consecutive Fibonacci numbers. On average, this leads to about 4% more comparisons to be executed, but it has the advantage that one only needs addition and subtraction to calculate the indices of the accessed array elements, while classical binary search needs bit-shift, division or multiplication, operations that were less common at the time Fibonacci search was first published. Fibonacci search has an average- and worst-case complexity of O(log n).
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 binary heap and binomial heap. Michael L. Fredman and Robert E. Tarjan developed Fibonacci heaps in 1984 and published them in a scientific journal in 1987. Fibonacci heaps are named after the Fibonacci numbers, which are used in their running time analysis.
Fibonacci numbers are also closely related to Lucas numbers in that they form a complementary pair of Lucas sequences and . Lucas numbers are also intimately connected with the golden ratio.
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
In mathematics, the Lucas sequences and are certain constant-recursive integer sequences that satisfy the recurrence relation
The Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985. + 1.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 m units is Fm
Knowledge of the Fibonacci sequence was expressed as early as Pingala (c. 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 m beats (Fm+1) is obtained by adding one [S] to the Fm cases and one [L] to the Fm−1 cases. Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD). However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), 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 mātrā-vṛttas [prosodic combinations].
Hemachandra (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 (1202) by Fibonacci.using it to calculate the growth of rabbit populations. Fibonacci considers the growth of a hypothetical, idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. Fibonacci posed the puzzle: how many pairs will there be in one year?
At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (that is, the number of pairs in month n − 2) plus the number of pairs alive last month (that is, n − 1). This is the nth Fibonacci number.
The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.
Joseph Schillinger (1895–1943) developed a system of composition which utilized Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.
Fibonacci sequences appear in biological settings,such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone, and the family tree of honeybees. Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers. Field daisies most often have petals in counts of Fibonacci numbers. In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.
Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.
A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogelin 1979. This has the form
where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j, which depends on r, the distance from the center. 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:
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, Fn, is the number of female ancestors, which is Fn−1, plus the number of male ancestors, which is Fn−2.This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
Luke Hutchison noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence. ), and at his parents' generation, his X chromosome came from a single parent (). 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 (). 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 (). Five great-great-grandparents contributed to the male descendant's X chromosome (), etc. (Note that 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.)A male individual has an X chromosome, which he received from his mother, and a Y chromosome, which he received from his father. The male counts as the "origin" of his own X chromosome (
The pathways of tubulins on intracellular microtubules arrange in patterns of 3, 5, 8 and 13.
The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):
These numbers also give the solution to certain enumerative problems.The most common is that of counting the number of compositions of 1s and 2s which sum to a given total n: there are Fn+1 ways to do this.
For example, if n = 5, then Fn+1 = F6 = 8 counts the eight compositions summing to 5:
1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2.
The Fibonacci numbers can be found in different ways among the set of binary strings, or equivalently, among the subsets of a given set.
The first 21 Fibonacci numbers Fn for n = 0, 1, 2, ..., 20 are:
The sequence can also be extended to negative index n using the re-arranged recurrence relation
which yields the sequence of "negafibonacci" numberssatisfying
Thus the bidirectional sequence is
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form solution. It has become known as "Binet's formula", though it was already known by Abraham de Moivre and Daniel Bernoulli:
is the golden ratio ( OEIS: A001622 ), and
Since , this formula can also be written as
To see this, φ and ψ are both solutions of the equationsnote that
so the powers of φ and ψ satisfy the Fibonacci recursion. In other words,
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 U0 = 0 and U1 = 1 then the resulting sequence Un 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.
Taking the starting values U0 and U1 to be arbitrary constants, a more general solution is:
for all n ≥ 0, the number Fn is the closest integer to . Therefore, it can be found by rounding, that is by the use of the nearest integer function:
or in terms of the floor function:
Similarly, if we already know that the number F > 1 is a Fibonacci number, we can determine its index within the sequence by
where can be computed using logarithms to other usual bases. For example, .
Johannes Kepler 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
This convergence holds regardless of the starting values, excluding 0 and 0, or any pair in the conjugate golden ratio, [ clarification needed ] This can be verified using Binet's formula. 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.
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 relationships yield Fibonacci numbers as the linear coefficients:
This equation can be proved by induction on n.
This expression is also true for n < 1 if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
which yields . The eigenvalues of the matrix A are and corresponding to the respective eigenvectors
As the initial value is
it follows that the nth term is
From this, the nth element in the Fibonacci series may be read off directly as a closed-form expression:
Equivalently, the same computation may performed by diagonalization of A through use of its eigendecomposition:
where and The closed-form expression for the nth element in the Fibonacci series is therefore given by
which again yields
The matrix A has a determinant of −1, and thus it is a 2×2 unimodular matrix.
This property can be understood in terms of the continued fraction representation for the golden ratio:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for φ, 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:
Taking the determinant of both sides of this equation yields Cassini's identity,
Moreover, since AnAm = An+m for any square matrix A, 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 n into n + 1),
In particular, with m = n,
These last two identities provide a way to compute Fibonacci numbers recursively in O(log(n)) arithmetic operations and in time O(M(n) log(n)), where M(n) is the time for the multiplication of two numbers of n digits. This matches the time for computing the nth 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).
The question may arise whether a positive integer x is a Fibonacci number. This is true if and only if one or both of or is a perfect square. This is because Binet's formula above can be rearranged to give
which allows one to find the position in the sequence of a given Fibonacci number.
This formula must return an integer for all n, so the radical expression must be an integer (otherwise the logarithm does not even return a rational number).
Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that Fn can be interpreted as the number of sequences of 1s and 2s that sum to n − 1. This can be taken as the definition of Fn, with the convention that F0 = 0, meaning no sum adds up to −1, and that F1 = 1, meaning the empty sum "adds up" to 0. Here, the order of the summand matters. For example, 1 + 2 and 2 + 1 are considered two different sums.
For example, the recurrence relation
or in words, the nth Fibonacci number is the sum of the previous two Fibonacci numbers, may be shown by dividing the Fn sums of 1s and 2s that add to n − 1 into two non-overlapping groups. One group contains those sums whose first term is 1 and the other those sums whose first term is 2. In the first group the remaining terms add to n − 2, so it has Fn-1 sums, and in the second group the remaining terms add to n − 3, so there are Fn−2 sums. So there are a total of Fn−1 + Fn−2 sums altogether, showing this is equal to Fn.
Similarly, it may be shown that the sum of the first Fibonacci numbers up to the nth is equal to the (n + 2)-nd Fibonacci number minus 1.In symbols:
This is done by dividing the sums adding to n + 1 in a different way, this time by the location of the first 2. Specifically, the first group consists of those sums that start with 2, the second group those that start 1 + 2, the third 1 + 1 + 2, and so on, until the last group, which consists of the single sum where only 1's are used. The number of sums in the first group is F(n), F(n − 1) in the second group, and so on, with 1 sum in the last group. So the total number of sums is F(n) + F(n − 1) + ... + F(1) + 1 and therefore this quantity is equal to F(n + 2).
A similar argument, grouping the sums by the position of the first 1 rather than the first 2, gives two more identities:
In words, the sum of the first Fibonacci numbers with odd index up to F2n−1 is the (2n)th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to F2n is the (2n + 1)th Fibonacci number minus 1.
A different trick may be used to prove
or in words, the sum of the squares of the first Fibonacci numbers up to Fn is the product of the nth and (n + 1)th Fibonacci numbers. In this case note that Fibonacci rectangle of size Fn by F(n + 1) can be decomposed into squares of size Fn, Fn−1, and so on to F1 = 1, from which the identity follows by comparing areas.
The sequence is also considered using the symbolic method. More precisely, this sequences corresponds to a specifiable combinatorial class. The specification of this sequence is . Indeed, as stated above, the -th Fibonacci numbes equals the number of way to partition using segments of size 1 or 2.
It follows that the ordinary generating function of the Fibonacci sequence, i.e. , is the complex function .
Numerous other identities can be derived using various methods. Some of the most noteworthy are:
Cassini's identity states that
Catalan's identity is a generalization:
where Ln is the n'th Lucas number. The last is an identity for doubling n; other identities of this type are
by Cassini's identity.
These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number.
Putting k = 2 in this formula, one gets again the formulas of the end of above section Matrix form.
The generating function of the Fibonacci sequence is the power series
This series is convergent for and its sum has a simple closed-form:
This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum:
Solving the equation
for s(x) results in the above closed form.
Setting x = 1/k, the closed form of the series becomes
In particular, if k is an integer greater than 1, then this series converges. Further setting k = 10m yields
for all positive integers m.
Some math puzzle-books present as curious the particular value that comes from m = 1, which is Similarly, m = 2 gives
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 nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,
No closed formula for the reciprocal Fibonacci constant
is known, but the number has been proved irrational by Richard André-Jeannin.
The Millin series gives the identity
which follows from the closed form for its partial sums as N tends to infinity:
Every third number of the sequence is even and more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property
Any three consecutive Fibonacci numbers are pairwise coprime, which means that, 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 Fp − 1, and if p is congruent to 2 or 3 (mod 5), then, p divides Fp + 1. The remaining case is that p = 5, and in this case p divides Fp.
These cases can be combined into a single formula, using the Legendre symbol:
The above formula can be used as a primality test in the sense that if
where the Legendre symbol has been replaced by the Jacobi symbol, 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 large—say a 500-bit number—then we can calculate Fm (mod n) efficiently using the matrix form. Thus
Here the matrix power Am is calculated using modular exponentiation, which can be adapted to matrices.
A Fibonacci prime is a Fibonacci number that is prime. The first few are:
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.
Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number.
The only nontrivial square 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, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.
With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). OEIS: A235383 .As a result, 8 and 144 (F6 and F12) 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 which is evaluated as follows:
If p is a prime number then
It is not known whether there exists a prime p such that
Such primes (if there are any) would be called Wall–Sun–Sun primes.
Also, if p ≠ 5 is an odd prime number then:
Example 1.p = 7, in this case p ≡ 3 (mod 4) and we have:
Example 2.p = 11, in this case p ≡ 3 (mod 4) and we have:
Example 3.p = 13, in this case p ≡ 1 (mod 4) and we have:
Example 4.p = 29, in this case p ≡ 1 (mod 4) and we have:
For odd n, all odd prime divisors of Fn are congruent to 1 modulo 4, implying that all odd divisors of Fn (as the products of odd prime divisors) are congruent to 1 modulo 4.
All known factors of Fibonacci numbers F(i) for all i < 50000 are collected at the relevant repositories.
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 periods OEIS: A001175 . 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 of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.
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 triple. 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 Fn, Fn+1, Fn+2 and Fn+3 can also be used to generate a Pythagorean triple in a different way:
Since Fn is asymptotic to , the number of digits in Fn is asymptotic to . 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 Fn is asymptotic to
The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. 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:
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.
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 residue (non-residue) is −1. Its value at zero is 0.
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ(n) or ϕ(n), and may also be called Euler's phi function. It can be defined more formally as the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.
Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as
In number theory, Euler's theorem states that if n and a are coprime positive integers, then
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem, at which time both of Fermat's theorems were already well known to mathematicians.
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.
In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/2, independently for different n. By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943…(sequence A078416 in the OEIS), a mathematical constant that was later named Viswanath's constant.
In number theory, the nth Pisano period, written π(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 periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774.
In number theory, the Carmichael function associates to every positive integer n a positive integer , defined as the smallest positive integer m such that
A Fibonacci word is a specific sequence of binary digits. The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.
In mathematics, the Fibonacci numbers form a sequence defined recursively by:
The following examples of generating functions are in the spirit of George Pólya, who advocated learning mathematics by doing and re-capitulating as many examples and proofs as possible. The purpose of this article is to present common ways of creating generating functions.
Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in n independent tosses of a fair coin, no run of k consecutive heads appears.
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:
The Leonardo numbers are a sequence of numbers given by the recurrence:
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.
The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word.
it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ...there are exactly Fm+1 of them. For example the 21 sequences when m = 7 are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)
Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)...
SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala [1969, 463–76], after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC
It is ironic that Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Édouard Lucas... attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber abaci
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