Prime number

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.

Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.

Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.

Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.

The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide

Hexadecimal

In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most the symbols "0"–"9" to represent values zero to nine, "A"–"F" to represent values ten to fifteen. Hexadecimal numerals are used by computer system designers and programmers, as they provide a more human-friendly representation of binary-coded values; each hexadecimal digit represents four binary digits known as a nibble, half a byte. For example, a single byte can have values ranging from 0000 0000 to 1111 1111 in binary form, which can be more conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is used to specify the radix. For example the decimal value 10,995 would be expressed in hexadecimal as 2AF316. In programming, a number of notations are used to support hexadecimal representation involving a prefix or suffix; the prefix 0x is used in C and related languages, which would denote this value by 0x2AF3. Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.

In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript can give the base explicitly: 15910 is decimal 159; some authors prefer a text subscript, such as 159decimal and 159hex, or 159h. In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen: In URIs, character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space character, ASCII code point 20 in hex, 32 in decimal. In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation ode, thus ’. In the Unicode standard, a character value is represented with U+ followed by the hex value, e.g. U+20AC is the Euro sign. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #: white, for example, is represented #FFFFFF.

CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33. Unix shells, AT&T assembly language and the C programming language use the prefix 0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits:'\x1B' represents the Esc control character. To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In MIME quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits prefixed by an equal to sign =, as in Espa=F1a to send "España". In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h: FFh or 05A3H; some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh instead of FFh Other assembly languages, Delphi, some versions of BASIC, GameMaker Language and Forth use $ as a prefix: $5A3.

Some assembly languages use the notation H'ABCD'. Fortran 95 uses Z'ABCD'. Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. For bit vector constants VHDL uses the notation x"5A3". Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant; the Smalltalk language uses the prefix 16r: 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#: 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC... Common Lisp uses the prefixes # 16r. Setting the variables *read-base* and *print-base* to 16 can be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers, thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16. MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3 BBC BASIC and Locomotive BASIC use & for hex.

TI-89 and 92 series uses a 0h prefix: 0h5A3 ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3. Binary and octal numbers can be specified similarly; the most common format for hexadecimal on IBM mainframes and midrange computers running the traditional OS's is X'5A3', is used in Assembler, PL/I, COBOL, JCL, scripts and other places. This format was common on

Natural number

In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.

A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.

Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.

Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a

Factorization

In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, is a factorization of the polynomial x2 – 4. Factorization is not considered meaningful within number systems possessing division, such as the real or complex numbers, since any x can be trivially written as × whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by ancient Greek mathematicians in the case of integers, they proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors.

Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact, exploited in the RSA cryptosystem to implement public-key cryptography. Polynomial factorization has been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique factorization into linear polynomials: this is a version of the fundamental theorem of algebra. In this case, the factorization can be done with root-finding algorithms; the case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing factorizations within the ring of polynomials with rational number coefficients.

A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals. Factorization may refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal entries equal to one, an upper triangular matrix U, a permutation matrix P. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique factorization into prime numbers, which are those integers which cannot be further factorized into the product of integers greater than one.

For computing the factorization of an integer n, one needs an algorithm for finding a divisor q of n or deciding that n is prime. When such a divisor is found, the repeated application of this algorithm to the factors q and n / q gives the complete factorization of n. For finding a divisor q of n, if any, it suffices to test all values of q such that 1 < q and q2 ≤ n. In fact, if r is a divisor of n such that r2 > n q = n / r is a divisor of n such that q2 ≤ n. If one tests the values of q in increasing order, the first divisor, found is a prime number, the cofactor r = n / q cannot have any divisor smaller than q. For getting the complete factorization, it suffices thus to continue the algorithm by searching a divisor of r, not smaller than q and not greater than √r. There is no need to test all values of q for applying the method. In principle, it suffices to test only prime divisors; this needs to have a table of prime numbers that may be generated for example with the sieve of Eratosthenes.

As the method of factorization does the same work as the sieve of Eratosthenes, it is more efficient to test for a divisor only those numbers for which it is not clear whether they are prime or not. One may proceed by testing 2, 3, 5, the numbers > 5, whose last digit is 1, 3, 7, 9 and the sum of digits is not a multiple of 3. This method is inefficient for larger integers. For example, Pierre de Fermat was unable to discover that the 6th Fermat number 1 + 2 2 5 = 1 + 2 32 = 4 294 967 297 is not a prime number. In fact, applying the above method would require more than 10000 divisions, for a number that has 10 decimal digits. There are more efficient factoring algorithms; however they remain inefficient, as, with the present state of the art, one cannot factorize with the more powerful computers, a number of 500 decimal digits, the product of two randomly chosen prime numbers. This insures the security of the RSA cryptosystem, used for secure internet communication. For fa

Roman numerals

The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, employ seven symbols, each with a fixed integer value, as follows: The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Arabic numerals; the original pattern for Roman numerals used the symbols I, V, X as simple tally marks. Each marker for 1 added a unit value up to 5, was added to to make the numbers from 6 to 9: I, II, III, IIII, V, VI, VII, VIII, VIIII, X; the numerals for 4 and 9 proved problematic, are replaced with IV and IX. This feature of Roman numerals is called subtractive notation; the numbers from 1 to 10 are expressed in Roman numerals as follows: I, II, III, IV, V, VI, VII, VIII, IX, X.

The system being decimal and hundreds follow the same underlying pattern. This is the key to understanding Roman numerals: Thus 10 to 100: X, XX, XXX, XL, L, LX, LXX, LXXX, XC, C. Note that 40 and 90 follow the same subtractive pattern as 4 and 9, avoiding the confusing XXXX. 100 to 1000: C, CC, CCC, CD, D, DC, DCC, DCCC, CM, M. Again - 400 and 900 follow the standard subtractive pattern, avoiding CCCC. In the absence of standard symbols for 5,000 and 10,000 the pattern breaks down at this point - in modern usage M is repeated up to three times; the Romans had several ways to indicate larger numbers, but for practical purposes Roman Numerals for numbers larger than 3,999 are if used nowadays, this suffices. M, MM, MMM. Many numbers include hundreds and tens; the Roman numeral system being decimal, each power of ten is added in descending sequence from left to right, as with Arabic numerals. For example: 39 = "Thirty nine" = XXXIX. 246 = "Two hundred and forty six" = CCXLVI. 421 = "Four hundred and twenty one" = CDXXI.

As each power of ten has its own notation there is no need for place keeping zeros, so "missing places" are ignored, as in Latin speech, thus: 160 = "One hundred and sixty" = CLX 207 = "Two hundred and seven" = CCVII 1066 = "A thousand and sixty six" = MLXVI. Roman numerals for large numbers are nowadays seen in the form of year numbers, as in these examples: 1776 = MDCCLXXVI. 1954 = MCMLIV 1990 = MCMXC. 2014 = MMXIV (the year of the games of the XXII Olympic Winter Games The current year is MMXIX. The "standard" forms described above reflect typical modern usage rather than an unchanging and universally accepted convention. Usage in ancient Rome varied and remained inconsistent in medieval times. There is still no official "binding" standard, which makes the elaborate "rules" used in some sources to distinguish between "correct" and "incorrect" forms problematic. "Classical" inscriptions not infrequently use IIII for "4" instead of IV. Other "non-subtractive" forms, such as VIIII for IX, are sometimes seen, although they are less common.

On the numbered gates to the colosseum, for instance, IV is systematically avoided in favour of IIII, but other "subtractives" apply, so that gate 44 is labelled XLIIII. Isaac Asimov speculates that the use of "IV", as the initial letters of "IVPITER" may have been felt to have been impious in this context. Clock faces that use Roman numerals show IIII for four o'clock but IX for nine o'clock, a practice that goes back to early clocks such as the Wells Cathedral clock of the late 14th century. However, this is far from universal: for example, the clock on the Palace of Westminster, Big Ben, uses a "normal" IV. XIIX or IIXX are sometimes used for "18" instead of XVIII; the Latin word for "eighteen" is rendered as the equivalent of "two less than twenty" which may be the source of this usage. The standard forms for 98 and 99 are XCVIII and XCIX, as described in the "decimal pattern" section above, but these numbers are rendered as IIC and IC originally from the Latin duodecentum and undecentum.

Sometimes V and L are not used, with instances such as IIIIII and XXXXXX rather than VI or LX. Most non-standard numerals other than those described above - such as VXL for 45, instead of the standard XLV are modern and may be due to error rather than being genuine variant usage. In the early years of the 20th century, different representations of 900 appeared in several inscribed dates. For instance, 1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, while on the north entrance to the Saint Louis Art Museum, 1903 is inscribed as MDCDIII rather than MCMIII. Although Roman numerals came to be written with letters

100 (number)

100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to describe the long hundred of six score or 120. 100 is the square of 10. The standard SI prefix for a hundred is "hecto-". 100 is the basis of percentages. 100 is the sum of the first nine prime numbers, as well as the sum of some pairs of prime numbers e.g. 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53. 100 is the sum of the cubes of the first four integers. This is related by Nicomachus's theorem to the fact that 100 equals the square of the sum of the first four integers: 100 = 102 = 2.26 + 62 = 100, thus 100 is a Leyland number.100 is an 18-gonal number. It is divisible by 25, the number of primes below it, it can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. It can be expressed as a sum of some of its divisors.

100 is a Harshad number in base 10, in base 4, in that base it is a self-descriptive number. There are 100 prime numbers whose digits are in ascending order. 100 is the smallest number. One hundred is the atomic number of fermium, an actinide and the first of the heavy metals that cannot be created through neutron bombardment. On the Celsius scale, 100 degrees is the boiling temperature of pure water at sea level; the Kármán line lies at an altitude of 100 kilometres above the Earth's sea level and is used to define the boundary between Earth's atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of the Jewish New Year. A religious Jew is expected to utter at least 100 blessings daily. In the Hindu book of the Mahabharata, the king Dhritarashtra had 100 sons known as the Kauravas; the United States Senate has 100 Senators. Most of the world's currencies are divided into 100 subunits; the 100 Euro banknotes feature a picture of a Rococo gateway on the obverse and a Baroque bridge on the reverse.

The U. S. hundred-dollar bill has Benjamin Franklin's portrait. S. bill in print. American savings bonds of $100 have Thomas Jefferson's portrait, while American $100 treasury bonds have Andrew Jackson's portrait. One hundred is also: The number of years in a century; the number of pounds in an American short hundredweight. In Greece, India and Nepal, 100 is the police telephone number. In Belgium, 100 is the firefighter telephone number. In United Kingdom, 100 is the operator telephone number; the HTTP status code indicating that the client should continue with its request. The 100 The age at which a person becomes a centenarian; the number of yards in an American football field. The number of runs required for a cricket batsman to score a significant milestone; the number of points required for a snooker player to score a century break, a significant milestone. The record number of points scored in one NBA game by a single player, set by Wilt Chamberlain of the Philadelphia Warriors on March 2, 1962.

1 vs. 100 AFI's 100 Years... Hundred Hundred Hundred Days Hundred Years' War List of highways numbered 100 Top 100 Greatest 100 Wells, D; the Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group.: 133 Chisholm, Hugh, ed.. "Hundred". Encyclopædia Britannica. Cambridge University Press. On the Number 100

Fibonacci number

In mathematics, the Fibonacci numbers 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, F 0 = 0, F 1 = 1, F n = F n − 1 + F n − 2, for n > 1. One has F2 = 1. In some books, in old ones, F0, the "0" is omitted, the Fibonacci sequence starts with F1 = F2 = 1; the beginning of the sequence is thus: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Fibonacci numbers are related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are named after Italian mathematician Leonardo of Pisa 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 numbers appear unexpectedly 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, graphs called Fibonacci cubes used for interconnecting parallel and distributed systems, they 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. Fibonacci numbers are closely related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. Lucas numbers are intimately connected with the golden ratio; the Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long syllables of 2 units duration, juxtaposed with short 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 + 1. Knowledge of the Fibonacci sequence was expressed as early as Pingala. Singh cites Pingala's cryptic formula misrau cha and scholars who interpret it in context as saying that the number of patterns for m beats is obtained by adding one to the Fm cases and one to the Fm−1 cases. Bharata Muni expresses knowledge of the sequence in the Natya Shastra. However, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala: Variations of two earlier meters... For example, for four, variations of meters of two three being mixed, five happens.... In this way, the process should be followed in all mātrā-vṛttas. Hemachandra 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 by Fibonacci. Using it to calculate the growth of rabbit populations. Fibonacci considers the growth of a hypothetical, idealized rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field. Fibonacci posed the puzzle: how many pairs will there be in one year? At the end of the first month, they mate. 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, making 5 pairs. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month; this is the nth Fibonacci number. The name "Fibonacci sequence" was first used by the 19th