1.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, 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. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the 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, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, 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, and 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, and 6 ones, and similarly for the number 4,622. A much later 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, 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, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the 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, 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 really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory

2.
Positional notation
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100

3.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0

4.
Portmanteau
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In linguistics, a portmanteau is defined as a single morph that represents two or more morphemes. A portmanteau also differs from a compound, which not involve the truncation of parts of the stems of the blended words. For instance, starfish is a compound, not a portmanteau, of star and fish, whereas a hypothetical portmanteau of star and fish might be stish. Humpty Dumpty explains the practice of combining words in various ways by telling Alice, for instance, take the two words fuming and furious. Make up your mind that you will say both words, but leave it unsettled which you will say first … if you have the rarest of gifts, in then-contemporary English, a portmanteau was a suitcase that opened into two equal sections. The etymology of the word is the French porte-manteau, from porter, to carry, in modern French, a porte-manteau is a clothes valet, a coat-tree or similar article of furniture for hanging up jackets, hats, umbrellas and the like. It has also used especially in Europe as a formal description for hat racks from the French words porter. An occasional synonym for portmanteau word is frankenword, an autological word exemplifying the phenomenon it describes, blending Frankenstein, many neologisms are examples of blends, but many blends have become part of the lexicon. In Punch in 1896, the word brunch was introduced as a portmanteau word, in 1964, the newly independent African republic of Tanganyika and Zanzibar chose the portmanteau word Tanzania as its name. Similarly Eurasia is a portmanteau of Europe and Asia, a scientific example is a liger, which is a cross between a male lion and a female tiger. Jeoportmanteau. is a category on the American television quiz show Jeopardy. The categorys name is itself a portmanteau of the words Jeopardy, responses in the category are portmanteaus constructed by fitting two words together. The term gerrymander has itself contributed to portmanteau terms bjelkemander and playmander, oxbridge is a common portmanteau for the UKs two oldest universities, those of Oxford and Cambridge. Many portmanteau words receive some use but do not appear in all dictionaries, for example, a spork is an eating utensil that is a combination of a spoon and a fork, and a skort is an item of clothing that is part skirt, part shorts. On the other hand, turducken, a made by inserting a chicken into a duck. Similarly, the word refudiate was first used by Sarah Palin when she misspoke, though initially a gaffe, the word was recognized as the New Oxford American Dictionarys Word of the Year in 2010. The business lexicon is replete with newly coined portmanteau words like permalance, advertainment, advertorial, infotainment, a company name may also be portmanteau as well as a product name. By contrast, the public, including the media, use portmanteaux to refer to their favorite pairings as a way to. giv people an essence of who they are within the same name and this is particularly seen in cases of fictional and real-life supercouples

5.
666 (number)
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666 is the natural number following 665 and preceding 667. Six hundred and sixty-six is called the number of the Beast in chapter 13 of the Book of Revelation, of the New Testament,666 is the sum of the first 36 natural numbers, and thus it is a triangular number. Notice that 36 =15 +21,15 and 21 are also triangular numbers, in base 10,666 is a repdigit and a Smith number. A prime reciprocal magic square based on 1/149 in base 10 has a total of 666. The prime factorization of 666 is 2 •32 •37, some manuscripts of the original Greek use the symbols χξϛ chi xi stigma, while other manuscripts spell out the number in words. In modern popular culture,666 has become one of the most widely recognized symbols for the Antichrist or, alternatively, the number 666 is purportedly used to invoke Satan. Earnest references to the number occur both among apocalypticist Christian groups and in explicitly anti-Christian subcultures, references in contemporary Western art or literature are, more likely than not, intentional references to the Beast symbolism. Such popular references are therefore too numerous to list and it is common to see the symbolic role of the integer 666 transferred to the digit sequence 6-6-6. Some people take the Satanic associations of 666 so seriously that they actively avoid things related to 666 or the digits 6-6-6, in some early biblical manuscripts, including Papyrus 115, the number is cited as 616. In the Bible,666 is the number of talents of gold Solomon collected each year, in the Bible,666 is the number of Adonikams descendants who return to Jerusalem and Judah from the Babylonian exile. In the Bible, there may be a latent reference to 666 in the name of the great sixth-century BC king of Babylon, commonly spelled Nebuchadnezzar, transliterating from the Book of Daniel, the name is Nebuchadrezzar or Nebuchadrezzur in the Book of Jeremiah. The number of name can be calculated, since Hebrew letters double as numbers. Nebuchadrezzar is 663, and Nebuchadrezzur,669, midway between the two variants is 666. If the mysteries of Jeremiah are to be related to those of Revelation, Nebuchadrezzar, using gematria, Neron Caesar transliterated from Greek into Hebrew produces the number 666. The Latin spelling of Nero Caesar transliterated into Hebrew produces the number 616, thus, in the Bible,666 may have been a coded reference to Nero the Roman Emperor from 55 to 68 AD. Is the magic sum, or sum of the constants of a six by six magic square. Is the sum of all the numbers on a roulette wheel, was a winning lottery number in the 1980 Pennsylvania Lottery scandal, in which equipment was tampered to favor a 4 or 6 as each of the three individual random digits. Was the original name of the Macintosh SevenDust computer virus that was discovered in 1998, the number is a frequent visual element of Aryan Brotherhood tattoos

6.
Six nines in pi
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A sequence of six 9s occurs in the decimal representation of π, starting at the 762nd decimal place. The earliest known mention of this occurs in Douglas Hofstadters 1985 book Metamagical Themas, where Hofstadter states I myself once learned 380 digits of π. This sequence of six nines is sometimes called the Feynman point, after physicist Richard Feynman, who has also been claimed to have stated this same idea in a lecture. It is not clear when, or even if, Feynman made such a statement, however, it is not mentioned in published biographies or in his autobiographies, π is conjectured to be, but not known to be, a normal number. For a randomly chosen normal number, the probability of a sequence of six digits occurring this early in the decimal representation is usually only about 0. 08%. However, if the sequence can overlap itself then the probability is less, the probability of six 9s in a row this early is about 10% less, or 0. 0686%. But the probability of a repetition of any digit six times starting in the first 762 digits is ten times greater, the early string of six 9s is also the first occurrence of four and five consecutive identical digits. The next appearance of four consecutive identical digits is of the digit 7 at position 1,589. Other patterns are possible, a repetition of a three times in the first three digits, or four times starting in the first ten digits, or five times in the first 100 digits. Each of these has about a 1% chance, so looking at repeats up to length 12, there is about a 10% chance of finding something as surprising as the six nines. From this point of view, the fact that we really do find a repeat of several digits after 762 digits is not really very surprising, the next sequence of six consecutive identical digits is again composed of 9s, starting at position 193,034. The next distinct sequence of six consecutive identical digits starts with the digit 8 at position 222,299, while strings of nine 9s next occur at position 590,331,982 and 640,787,382. The positions of the first occurrences of 9, alone and in strings of 2,3,4,5,6,7,8, and 9 consecutive 9s, are 5,44,762,762,762,762,1,722,776,36,356,642, and 564,665,206, respectively. The first 1001 digits of π, showing consecutive runs of three or more digits, including the consecutive six 9s underlined and coloured red, are as follows,0.999,9 Mathematical coincidence Repdigit Ramanujans constant Feynman Point Mathworld Article — From the Mathworld project

7.
Christian
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A Christian is a person who follows or adheres to Christianity, an Abrahamic, monotheistic religion based on the life and teachings of Jesus Christ. Christian derives from the Koine Greek word Christós, a translation of the Biblical Hebrew term mashiach, while there are diverse interpretations of Christianity which sometimes conflict, they are united in believing that Jesus has a unique significance. The term Christian is also used as an adjective to describe anything associated with Christianity, or in a sense all that is noble, and good. According to a 2011 Pew Research Center survey, there were 2.2 billion Christians around the world in 2010, by 2050, the Christian population is expected to exceed 3 billion. According to a 2012 Pew Research Center survey Christianity will remain the worlds largest religion in 2050, about half of all Christians worldwide are Catholic, while more than a third are Protestant. Orthodox communions comprise 12% of the worlds Christians, other Christian groups make up the remainder. Christians make up the majority of the population in 158 countries and territories,280 million Christian live as a minority. In the Greek Septuagint, christos was used to translate the Hebrew מָשִׁיחַ, in other European languages, equivalent words to Christian are likewise derived from the Greek, such as Chrétien in French and Cristiano in Spanish. The second mention of the term follows in Acts 26,28, where Herod Agrippa II replied to Paul the Apostle, Then Agrippa said unto Paul, Almost thou persuadest me to be a Christian. The third and final New Testament reference to the term is in 1 Peter 4,16, which believers, Yet if as a Christian, let him not be ashamed. The city of Antioch, where someone gave them the name Christians, had a reputation for coming up with such nicknames, in the Annals he relates that by vulgar appellation commonly called Christians and identifies Christians as Neros scapegoats for the Great Fire of Rome. Another term for Christians which appears in the New Testament is Nazarenes which is used by the Jewish lawyer Tertullus in Acts 24, the Hebrew equivalent of Nazarenes, Notzrim, occurs in the Babylonian Talmud, and is still the modern Israeli Hebrew term for Christian. A wide range of beliefs and practices is found across the world among those who call themselves Christian, denominations and sects disagree on a common definition of Christianity. Most Baptists and fundamentalists, for example, would not acknowledge Mormonism or Christian Science as Christian, in fact, the nearly 77 percent of Americans who self-identify as Christian are a diverse pluribus of Christianities that are far from any collective unity. The identification of Jesus as the Messiah is not accepted by Judaism, the term for a Christian in Hebrew is נוּצְרי, a Talmudic term originally derived from the fact that Jesus came from the Galilean village of Nazareth, today in northern Israel. Adherents of Messianic Judaism are referred to in modern Hebrew as יְהוּדִים מָשִׁיחַיים, the term Nasara rose to prominence in July 2014, after the Fall of Mosul to the terrorist organization Islamic State of Iraq and the Levant. The nun or ن— the first letter of Nasara—was spray-painted on the property of Christians ejected from the city, where there is a distinction, Nasrani refers to people from a Christian culture and Masihi is used by Christians themselves for those with a religious faith in Jesus. In some countries Nasrani tends to be used generically for non-Muslim Western foreigners, another Arabic word sometimes used for Christians, particularly in a political context, is Ṣalībī from ṣalīb which refers to Crusaders and has negative connotations

8.
Eschatology
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Eschatology /ˌɛskəˈtɒlədʒi/ is a part of theology concerned with the final events of history, or the ultimate destiny of humanity. This concept is referred to as the end of the world or end time. The word arises from the Greek ἔσχατος eschatos meaning last and -logy meaning the study of, the Oxford English Dictionary defines eschatology as The part of theology concerned with death, judgment, and the final destiny of the soul and of humankind. In the context of mysticism, the phrase refers metaphorically to the end of ordinary reality, in many religions it is taught as an existing future event prophesied in sacred texts or folklore. More broadly, eschatology may encompass related concepts such as the Messiah or Messianic Age, the end time, history is often divided into ages, which are time periods each with certain commonalities. One age comes to an end and a new age or world to come, where different realities are present, begins. When such transitions from one age to another are the subject of eschatological discussion, the phrase, end of the world, is replaced by end of the age, end of an era, or end of life as we know it. Much apocalyptic fiction does not deal with the end of time but rather with the end of a period of time, the end of life as it is now. It is usually a crisis that brings an end to current reality and ushers in a new way of living, thinking, or being. This crisis may take the form of the intervention of a deity in history, a war, eschatologies vary as to their degree of optimism or pessimism about the future. In some eschatologies, conditions are better for some and worse for others, e. g. heaven, in Baháí belief, creation has neither a beginning nor an end. Instead, the eschatology of other religions is viewed as symbolic, in Baháí belief, human time is marked by a series of progressive revelations in which successive messengers or prophets come from God. In this view, the heaven and hell are seen as symbolic terms for the persons spiritual progress. In Baháí belief, the coming of Baháulláh, the founder of the Baháí Faith, signals the fulfilment of previous eschatological expectations of Islam, Christianity and other major religions. Christian eschatology is concerned with death, a state, Heaven, hell, the return of Jesus. Eschatological passages are found in places, esp. Isaiah, Daniel, Ezekiel, Matthew 24, The Sheep and the Goats, and the Book of Revelation, the Second Coming of Christ is the central event in Christian eschatology. Most Christians believe that death and suffering will continue to exist until Christs return, there are, however, various views concerning the order and significance of other eschatological events

9.
Number of the Beast
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The Number of the Beast is a term in the Book of Revelation, of the New Testament, that is associated with the Beast of Revelation in chapter 13. In most manuscripts of the New Testament and in English translations of the Bible, in critical editions of the Greek text, such as the Novum Testamentum Graece, it is noted that 616 is a variant. In the oldest preserved manuscript as of 2017, as well as ancient sources like Codex Ephraemi Rescriptus. The Number of the beast is described in the passage of Revelation 13, the actual number is only mentioned once, in verse 18. Possible translations include not only to count, to reckon but also to vote or to decide, in the Textus Receptus, derived from Byzantine text-type manuscripts, the number 666 is represented by the final 3 letters χξϛ. 17καὶ ἵνα μή τις δύνηται ἀγοράσαι ἢ πωλῆσαι εἰ μὴ ὁ ἔχων τὸ χάραγμα, 18Ὧδε ἡ σοφία ἐστίν· ὁ ἔχων τὸν νοῦν ψηφισάτω τὸν ἀριθμὸν τοῦ θηρίου· ἀριθμὸς γὰρ ἀνθρώπου ἐστί· καὶ ὁ ἀριθμὸς αὐτοῦ χξϛʹ. The last letter of the Greek alphabet is not the equivalent of the English letter Z, the Greek letter stigma ligature represents the number 6. 18ὧδε ἡ σοφία ἐστίν· ὁ ἔχων νοῦν ψηφισάτω τὸν ἀριθμὸν τοῦ θηρίου, irenaeus knew about the 616 reading, but did not adopt it. In the 380s, correcting the existing Latin-language version of the New Testament, around 2005, a fragment from Papyrus 115, taken from the Oxyrhynchus site, was discovered at the Oxford Universitys Ashmolean Museum. It gave the number as 616 χιϛʹ. This fragment is the oldest manuscript of Revelation 13 found as of 2017, the age of a manuscript is not an indicator of the date of its writing but refers to how old the physical material is. All original biblical manuscripts are non-existent today, as they were held and copied onto new materials, eventually the originals fell apart, leaving fragments for a period and then only the copies. So the oldest texts might actually be found among the newest copies, Codex Ephraemi Rescriptus, known before the P115 finding but dating to after it, has 616 written in full, ἑξακόσιοι δέκα ἕξ, hexakosioi deka hex. Papyrus 115 and Ephraemi Rescriptus have led scholars to regard 616 as the original number of the beast. Associating the number of the beast as the duration of the beast’s reign Corresponding symbolism for the Antichrist, in Greek isopsephy and Hebrew gematria, every letter has a corresponding numeric value. Summing these numbers gives a value to a word or name. The use of isopsephy to calculate the number of the beast is used in many of the below interpretations, preterist theologians typically support the numerical interpretation that 666 is the equivalent of the name and title, Nero Caesar. A manner of speaking against the emperor without the Roman authorities knowing, also Nero Caesar in the Hebrew alphabet is נרון קסר NRON QSR, which when used as numbers represent 5020065010060200, which add to 666

10.
777 (number)
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The number 777 is significant in various religious and political contexts. According to the Bible, Lamech, the father of Noah lived for 777 years, the numbers 3 and 7 are considered both perfect numbers under Hebrew tradition. Christian denominations consider seven to be a number because Genesis says that God rested on the 7th day. Because God rested on the 7th day, that is the reason for the observance of the Hebrew Sabbath on the last day of the week, the 7th day of the week is indicated on the Hebrew Lunar calendar containing 13 months of four weeks each. According to the American publication, the Orthodox Study Bible,777 represents the threefold perfection of the Trinity, the Watcher Community uses it as a reference to the 7 angels,7 trumpets, and 7 bowls of wrath from the Book of Revelation. 777 is also found in the title of the book 777, in other traditions and teachings, seven is seen as the perfect number that holds creation and the universe together. Religious or mythological cosmology refers to seven heavens, ancient Indian spiritual texts detail seven chakras, the Afrikaner Resistance Movement, a Boer-nationalist movement in South Africa, utilized the number 777 as part of their emblem. The number refers to a triumph of Gods number 7 over the Devils number 666, on the AWB flag, the numbers are arranged in a triskelion shape, resembling the Nazi hakenkreuz. In Unixs chmod, the value 777 grants all file access permissions to all user types, boeing, the airplane manufacturing giant, released the airplane in June 1995. The family of 777s include the 777-200, 777-200ER, the 777-300, the 777-200LR Worldliner, the 777-300ER,777 is used on most slot machines in the United States to identify a jackpot

11.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing

12.
Radix
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In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for base in the arithmetical sense. In the system with radix 13, for example, a string of such as 398 denotes the number 3 ×132 +9 ×131 +8 ×130. More generally, in a system with radix b, a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, commonly used numeral systems include, For a larger list, see List of numeral systems. The octal and hexadecimal systems are used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, a similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two. However, other systems are possible, e. g. golden ratio base. Base Radix economy Non-standard positional numeral systems Base Convert, a floating-point base calculator MathWorld entry on base

13.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

14.
Exponentiation
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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5

15.
Achilles number
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An Achilles number is a number that is powerful but not a perfect power. A positive integer n is a number if, for every prime factor p of n. In other words, every prime factor appears at least squared in the factorization, however, not all powerful numbers are Achilles numbers, only those that cannot be represented as mk, where m and k are positive integers greater than 1. Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers. A number n = p1a1 p2a2 … pkak is powerful if min ≥2, if in addition gcd =1 the number is an Achilles number. The smallest pair of consecutive Achilles numbers is,5425069447 =73 ×412 ×9725425069448 =23 ×260412108 is a powerful number and its prime factorization is 22 ·33, and thus its prime factors are 2 and 3. Both 22 =4 and 32 =9 are divisors of 108, however,108 cannot be represented as mk, where m and k are positive integers greater than 1, so 108 is an Achilles number. 360 is not an Achilles number because it is not powerful, one of its prime factors is 5 but 360 is not divisible by 52 =25. Finally,784 is not an Achilles number and it is a powerful number, because not only are 2 and 7 its only prime factors, but also 22 =4 and 72 =49 are divisors of it. Nonetheless, it is a power,784 =24 ⋅72 =2 ⋅72 =2 =282. So it is not an Achilles number

16.
Power of two
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In mathematics, a power of two means a number of the form 2n where n is an integer, i. e. the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to values, so we have 1,2. Because two is the base of the numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100…000 or 0. 00…001, just like a power of ten in the decimal system, a word, interpreted as an unsigned integer, can represent values from 0 to 2n −1 inclusively. Corresponding signed integer values can be positive, negative and zero, either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this show up frequently in computer software. For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees at any time. Powers of two are used to measure computer memory. A byte is now considered eight bits (an octet, resulting in the possibility of 256 values, the prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024. However, in general, the term kilo has been used in the International System of Units to mean 1,000, binary prefixes have been standardized, such as kibi meaning 1,024. Nearly all processor registers have sizes that are powers of two,32 or 64 being most common, powers of two occur in a range of other places as well. For many disk drives, at least one of the size, number of sectors per track. The logical block size is almost always a power of two. Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of two or three powers of two, or powers of two minus one. For example,640 =512 +128 =128 ×5, put another way, they have fairly regular bit patterns. A prime number that is one less than a power of two is called a Mersenne prime, for example, the prime number 31 is a Mersenne prime because it is 1 less than 32. Similarly, a number that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational, the numbers that can be represented as sums of consecutive positive integers are called polite numbers, they are exactly the numbers that are not powers of two

17.
Power of 10
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In mathematics, a power of 10 is any of the integer powers of the number ten, in other words, ten multiplied by itself a certain number of times. By definition, the one is a power of ten. The first few powers of ten are,1,10,100,1000,10000,100000,1000000,10000000. In decimal notation the nth power of ten is written as 1 followed by n zeroes and it can also be written as 10n or as 1En in E notation. See order of magnitude and orders of magnitude for named powers of ten, there are two conventions for naming positive powers of ten, called the long and short scales. Where a power of ten has different names in the two conventions, the long scale namme is shown in brackets, googolplex, a much larger power of ten, was also introduced in that book. Scientific notation is a way of writing numbers of very large, a number written in scientific notation has a significand multiplied by a power of ten. Sometimes written in the form, m × 10n Or more compactly as, where n is positive, this indicates the number zeros after the number, and where the n is negative, this indicates the number of decimal places before the number. As an example,105 =100,000 10−5 =0.00001 The notation of mEn, known as E notation, is used in programming, spreadsheets and databases. Power of two SI prefix Cosmic View, inspiration for the film Powers of Ten Video Powers of Ten, US Public Broadcasting Service, made by Charles and Ray Eames. Starting at a picnic by the lakeside in Chicago, this film transports the viewer to the edges of the universe. Every ten seconds we view the point from ten times farther out until our own galaxy is visible only as a speck of light among many others. Returning to Earth with breathtaking speed, we move inward - into the hand of the sleeping picnicker - with ten times more magnification every ten seconds and our journey ends inside a proton of a carbon atom within a DNA molecule in a white blood cell

18.
Square number
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In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number

19.
Cube (algebra)
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In arithmetic and algebra, the cube of a number n is its third power, the result of the number multiplied by itself twice, n3 = n × n × n. It is also the number multiplied by its square, n3 = n × n2 and this is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n and it determines the side of the cube of a given volume. It is also n raised to the one-third power, both cube and cube root are odd functions,3 = −. The cube of a number or any other mathematical expression is denoted by a superscript 3, a cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The perfect cubes up to 603 are, Geometrically speaking, an integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger. For example,27 small cubes can be arranged into one larger one with the appearance of a Rubiks Cube, the difference between the cubes of consecutive integers can be expressed as follows, n3 −3 = 3n +1. There is no minimum perfect cube, since the cube of an integer is negative. For example, −4 × −4 × −4 = −64, unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25,75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6, some cube numbers are also square numbers, for example,64 is a square number and a cube number. This happens if and only if the number is a perfect sixth power, the last digits of each 3rd power are, It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1,8 or 9. That is their values modulo 9 may be only −1,1 and 0, every positive integer can be written as the sum of nine positive cubes. The equation x3 + y3 = z3 has no solutions in integers. In fact, it has none in Eisenstein integers, both of these statements are also true for the equation x3 + y3 = 3z3. The sum of the first n cubes is the nth triangle number squared,13 +23 + ⋯ + n 3 =2 =2. Proofs Charles Wheatstone gives a simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. Indeed, he begins by giving the identity n 3 = + + + ⋯ + ⏟ n consecutive odd numbers, kanim provides a purely visual proof, Benjamin & Orrison provide two additional proofs, and Nelsen gives seven geometric proofs

20.
Perfect power
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In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m >1, in this case, n may be called a perfect kth power. If k =2 or k =3, then n is called a square or perfect cube. Sometimes 1 is also considered a perfect power. The sum of the reciprocals of the perfect powers p without duplicates is, ∑ p 1 p = ∑ k =2 ∞ μ ≈0.874464368 … where μ is the Möbius function and this is sometimes known as the Goldbach-Euler theorem. Detecting whether or not a natural number n is a perfect power may be accomplished in many different ways. One of the simplest such methods is to all possible values for k across each of the divisors of n. This method can immediately be simplified by considering only prime values of k. This is because if n = m k for a composite k = a p p is prime. Because of this result, the value of k must necessarily be prime. As an example, consider n = 296·360·724, since gcd =12, n is a perfect 12th power. In 2002 Romanian mathematician Preda Mihăilescu proved that the pair of consecutive perfect powers is 23 =8 and 32 =9. Pillais conjecture states that for any positive integer k there are only a finite number of pairs of perfect powers whose difference is k. As an alternate way to perfect powers, the recursive approach has yet to be found useful. It is based on the observation that the difference between ab and b where a > b may not be constant, but if you take the difference of differences, b times. For example,94 =6561, and 104 is 10000, the difference between 84 and 94 is 2465, meaning the difference of differences is 974. A step further and you have 204, one step further, and you have 24, which is equal to 4. One step further and collating this key row from progressively larger exponents yields a similar to Pascals

21.
Powerful number
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A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a number is the product of a square and a cube, that is, a number m of the form m = a2b3. Powerful numbers are known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful, in the other direction, suppose that m is powerful, with prime factorization m = ∏ p i α i, where each αi ≥2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi - γi. Then, all values βi are nonnegative integers, and all values γi are either zero or three, so m = =23 supplies the desired representation of m as a product of a square. Informally, given the prime factorization of m, take b to be the product of the factors of m that have an odd exponent. Because m is powerful, each prime factor with an odd exponent has an exponent that is at least 3, in addition, each prime factor of m/b3 has an even exponent, so m/b3 is a perfect square, so call this a2, then m = a2b3. The representation m = a2b3 calculated in this way has the property that b is squarefree, the sum of the reciprocals of the powerful numbers converges. More generally, the sum of the reciprocals of the sth powers of the numbers is equal to ζ ζ ζ whenever it converges. Let k denote the number of numbers in the interval. Then k is proportional to the root of x. More precisely, c x 1 /2 −3 x 1 /3 ≤ k ≤ c x 1 /2, c = ζ / ζ =2.173 …, the two smallest consecutive powerful numbers are 8 and 9. However, one of the two numbers in a pair formed in this way must be a square. According to Guy, Erdős has asked whether there are many pairs of consecutive powerful numbers such as in which neither number in the pair is a square. Jaroslaw Wroblewski showed that there are indeed infinitely many such pairs by showing that 33c2 +1 = 73d2 has infinitely many solutions and it is a conjecture of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers. Any odd number is a difference of two squares,2 = k2 + 2k +1, so 2 − k2 = 2k +1. Similarly, any multiple of four is a difference of the squares of two numbers that differ by two,2 − k2 = 4k +4

22.
Recursion
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Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic, the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines a number of instances, it is often done in such a way that no loop or infinite chain of references can occur. The ancestors of ones ancestors are also ones ancestors, the Fibonacci sequence is a classic example of recursion, Fib =0 as base case 1, Fib =1 as base case 2, For all integers n >1, Fib, = Fib + Fib. Many mathematical axioms are based upon recursive rules, for example, the formal definition of the natural numbers by the Peano axioms can be described as,0 is a natural number, and each natural number has a successor, which is also a natural number. By this base case and recursive rule, one can generate the set of all natural numbers, recursively defined mathematical objects include functions, sets, and especially fractals. There are various more tongue-in-cheek definitions of recursion, see recursive humor, Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be recursive, to understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, the running of a procedure involves actually following the rules and performing the steps. An analogy, a procedure is like a recipe, running a procedure is like actually preparing the meal. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure. For instance, a recipe might refer to cooking vegetables, which is another procedure that in turn requires heating water, for this reason recursive definitions are very rare in everyday situations. An example could be the procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point, If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively, if every trial fails by reaching only dead ends, return on the path led to this branching point. Whether this actually defines a terminating procedure depends on the nature of the maze, in any case, executing the procedure requires carefully recording all currently explored branching points, and which of their branches have already been exhaustively tried. This can be understood in terms of a definition of a syntactic category. A sentence can have a structure in which what follows the verb is another sentence, Dorothy thinks witches are dangerous, so a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, and optionally another sentence

23.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are 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. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, 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. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. 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 month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. 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 and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that 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, 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, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently