1.
Perfect Number (film)
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Perfect Number is a 2012 South Korean mystery-drama film directed by Bang Eun-jin. Cho Jin-woong received a Best Supporting Actor nomination at the 49th Baeksang Arts Awards in 2013, kim Seok-go showed a lot of promise as a brilliant mathematician when he was in school, resolutely focused on his studies rather than on friends throughout his childhood. Now in his 30s, hes a high school math teacher. Seok-go is solemn and introverted, and his exchanges with Baek Hwa-sun. When Hwa-suns ex-husband mercilessly beats Hwa-sun and her niece, Hwa-sun kills him, Seok-go overhears the fight from his house next door and decides to cover up the killing and protect her from the police. He uses his genius in planning the perfect alibi for her. However, the detective in charge, Jo Min-beom, believes that Hwa-sun is guilty and follows his intuition despite the lack of evidence
2.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
3.
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
4.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
5.
Divisor function
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In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the function, it counts the number of divisors of an integer. It appears in a number of identities, including relationships on the Riemann zeta function. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities, a related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. The sum of divisors function σx, for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as σ x = ∑ d ∣ n d x, the notations d, ν and τ are also used to denote σ0, or the number-of-divisors function. When x is 1, the function is called the function or sum-of-divisors function. The aliquot sum s of n is the sum of the proper divisors, and equals σ1 − n, the cases x =2 to 5 are listed in A001157 − A001160, x =6 to 24 are listed in A013954 − A013972. For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and σ0 is even, for an integer, one divisor is not paired with a distinct divisor. Similarly, the number σ1 is odd if and only if n is a square or twice a square. For a prime p, σ0 =2 σ0 = n +1 σ1 = p +1 because by definition. Also, where pn# denotes the primorial, σ0 =2 n since n prime factors allow a sequence of binary selection from n terms for each proper divisor formed, clearly,1 < σ0 < n and σ > n for all n >2. The divisor function is multiplicative, but not completely multiplicative and it follows that d is, σ0 = ∏ i =1 r. For example, if n is 24, there are two factors, noting that 24 is the product of 23×31, a1 is 3. Thus we can calculate σ0 as so, σ0 = ∏ i =12 = =4 ⋅2 =8, the eight divisors counted by this formula are 1,2,4,8,3,6,12, and 24. Here s denotes the sum of the divisors of n, that is. This function is the one used to perfect numbers which are the n for which s = n. If s > n then n is an abundant number and if s < n then n is a deficient number
6.
Euclid's Elements
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Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
7.
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
8.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
9.
6 (number)
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6 is the natural number following 5 and preceding 7. The SI prefix for 10006 is exa-, and for its reciprocal atto-,6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number, its proper divisors are 1,2 and 3, since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and S -perfect number. As a perfect number,6 is related to the Mersenne prime 3,6 is the only even perfect number that is not the sum of successive odd cubes. As a perfect number,6 is the root of the 6-aliquot tree, and is itself the sum of only one number. Six is the number that is both the sum and the product of three consecutive positive numbers. Unrelated to 6 being a number, a Golomb ruler of length 6 is a perfect ruler. Six is the first discrete biprime and the first member of the discrete biprime family, Six is the smallest natural number that can be written as the sum of two positive rational cubes which are not integers,6 =3 +3. Six is a perfect number, a harmonic divisor number and a superior highly composite number. The next superior highly composite number is 12,5 and 6 form a Ruth-Aaron pair under either definition. There are no Graeco-Latin squares with order 6, if n is a natural number that is not 2 or 6, then there is a Graeco-Latin square with order n. The smallest non-abelian group is the symmetric group S3 which has 3, s6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of mathematical objects such as the S Steiner system, the projective plane of order 4. This can also be expressed category theoretically, consider the category whose objects are the n element sets and this category has a non-trivial functor to itself only for n =6. 6 similar coins can be arranged around a central coin of the radius so that each coin makes contact with the central one. This makes 6 the answer to the kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the lattice in which each circle touches just six others. 6 is the largest of the four all-Harshad numbers, a six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane
10.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0
11.
Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
12.
Philo
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Philo of Alexandria, also called Philo Judaeus, was a Hellenistic Jewish philosopher who lived in Alexandria, in the Roman province of Egypt. Philo used philosophical allegory to attempt to fuse and harmonize Greek philosophy with Jewish philosophy and his method followed the practices of both Jewish exegesis and Stoic philosophy. His allegorical exegesis was important for several Christian Church Fathers, and he believed that literal interpretations of the Hebrew Bible would stifle humanitys view and perception of a God too complex and marvelous to be understood in literal human terms. Some scholars hold that his concept of the Logos as Gods creative principle influenced early Christology, other scholars, however, deny direct influence but say both Philo and Early Christianity borrow from a common source. The few biographical details known about Philo are found in his own works, the only event in his life that can be decisively dated is his participation in the embassy to Rome in 40 CE. He represented the Alexandrian Jews before Roman Emperor Caligula because of civil strife between the Alexandrian Jewish and Greek communities, Philo was probably born with the name Julius Philo. His ancestors and family were contemporaries to the rule of the Ptolemaic dynasty, although the names of his parents are unknown, Philo came from a family which was noble, honourable and wealthy. It was either his father or paternal grandfather who was granted Roman citizenship from Roman dictator Gaius Julius Caesar, Philo had two brothers, Alexander the Alabarch and Lysimachus. His ancestors and family had social ties and connections to the priesthood in Judea, the Hasmonean Dynasty, the Herodian Dynasty, Philo visited the Temple in Jerusalem at least once in his lifetime. Philo would have been a contemporary to Jesus of Nazareth and his Apostles, Philo along with his brothers received a thorough education. Philo, through his brother Alexander, had two nephews Tiberius Julius Alexander and Marcus Julius Alexander, Marcus Julius Alexander was the first husband of the Herodian Princess Berenice. We find a reference to Philo by the 1st-century Jewish historian Josephus. In Antiquities of the Jews, Josephus tells of Philos selection by the Alexandrian Jewish community as their principal representative before the Roman emperor Gaius Caligula. He says that Philo agreed to represent the Alexandrian Jews in regard to civil disorder that had developed between the Jews and the Greeks in Alexandria, Egypt. Josephus also tells us that Philo was skilled in philosophy, according to Josephus, Philo and the larger Jewish community refused to treat the emperor as a god, to erect statues in honor of the emperor, and to build altars and temples to the emperor. Josephus says Philo believed that God actively supported this refusal, many of these severe things were said by Apion, by which he hoped to provoke Gaius to anger at the Jews, as he was likely to be. Our remaining information about Philo is based upon his own writings, Philo himself claims in his Embassy to Gaius to have been part of an embassy sent by the Alexandrian Jews to the Roman Emperor Caligula. Philo says he was carrying a petition which described the sufferings of the Alexandrian Jews, Philo says he was regarded by his people as having unusual prudence, due to his age, education, and knowledge
13.
Origen
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Origen, or Origen Adamantius, was a Greek scholar, ascetic, and early Christian theologian who was born and spent the first half of his career in Alexandria. He was a writer in multiple branches of theology, including textual criticism, biblical exegesis and hermeneutics, philosophical theology, preaching. He was anathematised at the Second Council of Constantinople and he was one of the most influential figures in early Christian asceticism. Origens Greek name Ōrigénēs probably means child of Horus and his nickname or cognomen Adamantios derives from Greek adámas, which means adamant, unalterable, unbreakable, unconquerable, diamond. He acquired it because of his ascetical practices. Origen was born in Alexandria to Christian parents and he was educated by his father, Leonides of Alexandria, who gave him a standard Hellenistic education, but also had him study the Christian scriptures. The name of his mother is unknown, in 202, Origens father was martyred in the outbreak of the persecution during the reign of Septimius Severus. A story reported by Eusebius has it that Origen wished to follow him in martyrdom, the death of Leonides left the family of nine impoverished when their property was confiscated. Many modern scholars, however, doubt that Clements school had been an ecclesiastical institution as Origens was. His fame and the number of his pupils increased rapidly, so that Bishop Demetrius of Alexandria, Origen, to be entirely independent, sold his library for a sum which netted him a daily income of 4 obols, on which he lived by exercising the utmost frugality. Teaching throughout the day, he devoted the greater part of the night to the study of the Bible, Eusebius reported that Origen, following Matthew 19,12 literally, castrated himself. The later church historian Philostorgius of Apamea, on the other hand, Eusebius story was accepted during the Middle Ages and was cited by Peter Abelard in his letters to Heloise. Edward Gibbon, in his History of the Decline and Fall of the Roman Empire, during the past century, scholars have often questioned this, surmising that this may have been a rumor circulated by his detractors. Henry Chadwick points out that, while the story may be true, it seems unlikely, however, many noted historians, such as Peter Brown and William Placher, continue to find no reason to deny the truth of Eusebius claims. But the school had far outgrown the strength of a man, the catechumens pressed eagerly for elementary instruction. Under these circumstances, Origen entrusted the teaching of the catechumens to Heraclas and his own interests became more and more centered in exegesis, and he accordingly studied Hebrew, though there is no certain knowledge concerning his instructor in that language. From about this period dates Origens acquaintance with Ambrose of Alexandria, later Ambrose, a man of wealth, made a formal agreement with Origen to promulgate his writings, and all the subsequent works of Origen were dedicated to Ambrose. In the following year, an uprising at Alexandria caused Caracalla to let his soldiers plunder the city, shut the schools
14.
Didymus the Blind
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Didymus the Blind was a Christian theologian in the Church of Alexandria, whose famous Catechetical School he led for about half a century. Despite his impaired vision, his memory was so powerful that he mastered dialectics and geometry, subjects whose study usually benefits appreciably from sight. Didymus wrote many works, Commentaries on all the Psalms, the Gospel of Matthew, the Gospel of John as Against the Arians, and On the Holy Spirit and he also wrote on Isaiah, Hosea, Zechariah, Job, and many other topics. Didymus’ biblical commentaries, which supposedly addressed nearly all the books of the Bible and his Catholic Letters are of dubious authenticity. He is probably the author of a treatise on the Holy Spirit that is extant in Latin translation and he was a loyal follower of Origen, and opposed Arian and Macedonian teachings. Such of his writings as survive show a knowledge of scripture. Although he became blind at the age of four, before he had learned to read, despite his blindness, Didymus excelled in scholarship because of his incredible memory. He found ways to help people to read, and experimented with carved wooden letters. Braille systems used by the blind today, Rufinus recounts that upon entering the service of the Church, Didymus became a teacher in the Church school, having been approved by Bishop Athanasius and other learned churchmen. It used to be assumed that this meant he was placed at the head of the Catechetical School of Alexandria which had flourished under Clement and Origen. However, it has long been questioned whether this still existed in Didymus time. According to Palladius, the 5th-century bishop and historian, Didymus remained a layman all his life, Jerome, generally spoke of Didymus not as the blind but as the Seeing, or the Seer, since although blind, his writings showed great insight into God. Jerome also wrote that Didymus surpassed all of his day in knowledge of the Scriptures and Socrates of Constantinople later called him the great bulwark of the true faith, Didymus was viewed as an orthodox Christian teacher and was greatly respected and admired up until at least 553. Several Oriental Orthodox Churches refer to him as St. Didymus the Blind, in 553 the Second Council of Constantinople condemned his works, along with those of Origen and Evagrius, but not his person. In the Third Council of Constantinople in 680, Didymus was again linked with, as a result of his condemnation, many of his works were not copied during the Middle Ages and were subsequently lost. Of his lost compositions we can gather a partial list from the citations of ancient authors which includes On Dogmas, On The Death of Young Children, Against the Arians, First Word, and others. According to Jerome, he produced a commentary on Origens First Principles which tried, ultimately unsuccessfully. According to Palladius, Didymus also authored a work on both the Old and New Testaments, mostly believed to be lost
15.
The City of God (book)
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The City of God Against the Pagans, often called The City of God, is a book of Christian philosophy written in Latin by Augustine of Hippo in the early 5th century AD. The sack of Rome by the Visigoths in 410 left Romans in a state of shock. He attempted to console Christians, writing even if the earthly rule of the Empire was imperiled. Christianity, he argued, should be concerned with the mystical, heavenly city, the book presents human history as a conflict between what Augustine calls the Earthly City and the City of God, a conflict that is destined to end in victory for the latter. The City of God is marked by people who forgot earthly pleasure to dedicate themselves to the truths of God. The Earthly City, on the hand, consists of people who have immersed themselves in the cares and pleasures of the present. Augustine’s thesis depicts the history of the world as universal warfare between God and the Devil and this metaphysical war is not limited by time but only by geography on Earth. Many Catholics consider Jacques-Bénigne Bossuets Discours sur lhistoire universelle or Speech of Universal History to be a second edition or continuation of The City of God and this book updates universal history according to Augustine’s thesis of universal war between those humans that follow God and those who follow the Devil. Holds that in her most benign Lord and Master can be found the key, the focal point, all of human life, whether individual or collective, shows itself to be a dramatic struggle between good and evil, between light and darkness. The Lord is the goal of history the focal point of the longings of history and of civilization, the center of the human race, the joy of every heart. Augustine provides a description of the contents of the work, However. Of these twelve books, the first four contain an account of the origin of these two city of God, and the city of the world. The second four treat of their history or progress, the third and last four, the book also explains good and bad things happen to righteous and wicked people alike, and it consoles the women violated in the recent calamity. Book II, a proof that because of the worship of the gods, Rome suffered the greatest calamity of all. Book III, a proof that the gods failed to save Rome numerous times in the past from worldly disasters. Book IV, a proof that the power and long duration of the Roman empire was due not to the pagan Gods, Book V, a refutation of the doctrine of fate and an explanation of the Christian doctrine of free will and its consistency with Gods omniscience. The book proves that Romes dominion was due to the virtue of the Romans and explains the true happiness of the Christian emperors, Book VI–X, A critique of pagan philosophy Book VI, a refutation of the assertion that the pagan gods are to be worshiped for eternal life. Augustine claimed that even the esteemed pagan theologist Varro held the gods in contempt, Book VII, a demonstration that eternal life is not granted by Janus, Jupiter, Saturn, and other select gods
16.
Pietro Cataldi
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Pietro Antonio Cataldi was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems and his work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to prove Euclids fifth postulate, Cataldi discovered the sixth and seventh primes later to acquire the designation Mersenne primes by 1588. Although Cataldi also claimed that p=23,29,31 and 37 all also generate Mersenne primes, oConnor, John J. Robertson, Edmund F. Pietro Cataldi, MacTutor History of Mathematics archive, University of St Andrews
17.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
18.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
19.
Marin Mersenne
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Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the father of acoustics. Mersenne, an ordained priest, had contacts in the scientific world and has been called the center of the world of science. Marin Mersenne was born of peasant parents near Oizé, Maine and he was educated at Le Mans and at the Jesuit College of La Flèche. On 17 July 1611, he joined the Minim Friars, and, after studying theology, between 1614 and 1618, he taught theology and philosophy at Nevers, but he returned to Paris and settled at the convent of LAnnonciade in 1620. There he studied mathematics and music and met with other kindred spirits such as René Descartes, Étienne Pascal, Pierre Petit, Gilles de Roberval and he corresponded with Giovanni Doni, Constantijn Huygens, Galileo Galilei, and other scholars in Italy, England and the Dutch Republic. He was a defender of Galileo, assisting him in translations of some of his mechanical works. For four years, Mersenne devoted himself entirely to philosophic and theological writing and it is sometimes incorrectly stated that he was a Jesuit. He was educated by Jesuits, but he never joined the Society of Jesus and he taught theology and philosophy at Nevers and Paris. He was not afraid to cause disputes among his friends in order to compare their views. In 1635 Mersenne met with Tommaso Campanella, but concluded that he could teach nothing in the sciences but still he has a good memory, Mersenne asked if René Descartes wanted Campanella to come to Holland to meet him, but Descartes declined. He visited Italy fifteen times, in 1640,1641 and 1645, in 1643–1644 Mersenne also corresponded with the German Socinian Marcin Ruar concerning the Copernican ideas of Pierre Gassendi, finding Ruar already a supporter of Gassendis position. Among his correspondents were Descartes, Galilei, Roberval, Pascal, Beeckman and he died September 1 through complications arising from a lung abscess. Some history scientists suggest he died for having drunk a huge quantity of water, along with Descartes. It was written as a commentary on the Book of Genesis, at first sight the book appears to be a collection of treatises on various miscellaneous topics. However Robert Lenoble has shown that the principle of unity in the work is a polemic against magical and divinatory arts, cabalism and he mentions Martin Del Rios Investigations into Magic and criticises Marsilio Ficino for claiming power for images and characters. He condemns astral magic and astrology and the anima mundi, a popular amongst Renaissance neo-platonists. Whilst allowing for an interpretation of the Cabala, he wholeheartedly condemned its magical application—particularly to angelology. He also criticises Pico della Mirandola, Cornelius Agrippa and Francesco Giorgio with Robert Fludd as his main target, Fludd responded with Sophia cum moria certamen, wherein Fludd admits his involvement with the Rosicrucians
20.
Ibn al-Haytham
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Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, and philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and he was the first to explain that vision occurs when light bounces on an object and then is directed to ones eyes. He spent most of his close to the court of the Fatimid Caliphate in Cairo and earned his living authoring various treatises. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called The Physicist and he is also sometimes called al-Baṣrī after his birthplace Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born c.965 in Basra, which was part of the Buyid emirate. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. Legend has it that after deciding the scheme was impractical and fearing the caliphs anger, during this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. Among his students were Sorkhab, a Persian from Semnan who was his student for three years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen. Alhazen made significant contributions to optics, number theory, geometry, astronomy, Alhazens work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text. A Latin translation of the Kitab al-Manazir was made probably in the twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration and he made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the known as Alhazens problem. Alhazen wrote as many as 200 books, although only 55 have survived, some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew, the crater Alhazen on the Moon is named in his honour, as was the asteroid 59239 Alhazen. In honour of Alhazen, the Aga Khan University named its Ophthalmology endowed chair as The Ibn-e-Haitham Associate Professor, Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10, 000-dinar banknote issued in 2003, and on 10-dinar notes from 1982. The 2015 International Year of Light celebrated the 1000th anniversary of the works on optics by Ibn Al-Haytham, Alhazens most famous work is his seven-volume treatise on optics Kitab al-Manazir, written from 1011 to 1021. Optics was translated into Latin by a scholar at the end of the 12th century or the beginning of the 13th century
21.
Bijection
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In mathematical terms, a bijective function f, X → Y is a one-to-one and onto mapping of a set X to a set Y. A bijection from the set X to the set Y has a function from Y to X. If X and Y are finite sets, then the existence of a means they have the same number of elements. For infinite sets the picture is complicated, leading to the concept of cardinal number. A bijective function from a set to itself is called a permutation. Bijective functions are essential to many areas of including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group. Satisfying properties and means that a bijection is a function with domain X and it is more common to see properties and written as a single statement, Every element of X is paired with exactly one element of Y. Functions which satisfy property are said to be onto Y and are called surjections, Functions which satisfy property are said to be one-to-one functions and are called injections. With this terminology, a bijection is a function which is both a surjection and an injection, or using words, a bijection is a function which is both one-to-one and onto. Consider the batting line-up of a baseball or cricket team, the set X will be the players on the team and the set Y will be the positions in the batting order The pairing is given by which player is in what position in this order. Property is satisfied since each player is somewhere in the list, property is satisfied since no player bats in two positions in the order. Property says that for each position in the order, there is some player batting in that position, in a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them all to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. The instructor was able to conclude there were just as many seats as there were students. For any set X, the identity function 1X, X → X, the function f, R → R, f = 2x +1 is bijective, since for each y there is a unique x = /2 such that f = y. In more generality, any linear function over the reals, f, R → R, f = ax + b is a bijection, each real number y is obtained from the real number x = /a. The function f, R →, given by f = arctan is bijective since each real x is paired with exactly one angle y in the interval so that tan = x
22.
Great Internet Mersenne Prime Search
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The Great Internet Mersenne Prime Search is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. The GIMPS project was founded by George Woltman, who wrote the software Prime95. Scott Kurowski wrote the PrimeNet Internet server that supports the research to demonstrate Entropia-distributed computing software, GIMPS is registered as Mersenne Research, Inc. Kurowski is Executive Vice President and board director of Mersenne Research Inc, GIMPS is said to be one of the first large scale distributed computing projects over the Internet for research purposes. The project has found a total of fifteen Mersenne primes as of January 2016, the largest known prime as of January 2016 is 274,207,281 −1. This prime was discovered on September 17,2015 by Curtis Cooper at the University of Central Missouri and they also have a trial division phase, used to rapidly eliminate Mersenne numbers with small factors which make up a large proportion of candidates. Pollards p -1 algorithm is used to search for larger factors. The project began in early January 1996, with a program ran on i386 computers. The name for the project was coined by Luther Welsh, one of its earlier searchers, within a few months, several dozen people had joined, and over a thousand by the end of the first year. Joel Armengaud, a participant, discovered the primality of M1,398,269 on November 13,1996, as of March 2013, GIMPS has a sustained aggregate throughput of approximately 137.023 TFLOP/s. In November 2012, GIMPS maintained 95 TFLOP/s, theoretically earning the GIMPS virtual computer a place among the TOP500 most powerful computer systems in the world. Also theoretically, in November 2012, the GIMPS held a rank of 330 in the TOP500, the preceding place was then held by an HP Cluster Platform 3000 BL460c G7 of Hewlett-Packard. As of November 2014 TOP500 results, these old GIMPS numbers would no longer make the list, previously, this was approximately 50 TFLOP/s in early 2010,30 TFLOP/s in mid-2008,20 TFLOP/s in mid-2006, and 14 TFLOP/s in early 2004. Third-party programs for testing Mersenne numbers, such as Mlucas and Glucas, also, GIMPS reserves the right to change this EULA without notice and with reasonable retroactive effect. All Mersenne primes are in the form Mq, where q is the exponent, the prime number itself is 2q −1, so the smallest prime number in this table is 21398269 −1. Mn is the rank of the Mersenne prime based on its exponent, furthermore,71,027,647 is the largest exponent below which all other exponents have been tested at least once, so some Mersenne numbers between the 48th and the 49th have yet to be tested. ^ ‡ The number M74207281 has 22,338,618 decimal digits, to help visualize the size of this number, a standard word processor layout would require 5,957 pages to display it. If one were to print it out using standard printer paper, single-sided, whenever a possible prime is reported to the server, it is verified first before it is announced
23.
Triangular number
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime
24.
Hexagonal number
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A hexagonal number is a figurate number. The formula for the nth hexagonal number h n =2 n 2 − n = n =2 n ×2. The first few numbers are,1,6,15,28,45,66,91,120,153,190,231,276,325,378,435,496,561,630,703,780,861,946. Every hexagonal number is a number, but only every other triangular number is a hexagonal number. Like a triangular number, the root in base 10 of a hexagonal number can only be 1,3,6. The digital root pattern, repeating every nine terms, is 166193139. Every even perfect number is hexagonal, given by the formula M p 2 p −1 = M p /2 = h /2 = h 2 p −1 where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal, for example, the 2nd hexagonal number is 2×3 =6, the 4th is 4×7 =28, the 16th is 16×31 =496, and the 64th is 64×127 =8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130, adrien-Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. Hexagonal numbers can be rearranged into rectangular numbers of n by. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages, to avoid ambiguity, hexagonal numbers are sometimes called cornered hexagonal numbers. One can efficiently test whether a positive x is an hexagonal number by computing n =8 x +1 +14. If n is an integer, then x is the nth hexagonal number, if n is not an integer, then x is not hexagonal. The nth number of the sequence can also be expressed by using Sigma notation as h n = ∑ i =0 n −1 where the empty sum is taken to be 0. Centered hexagonal number Mathworld entry on Hexagonal Number
25.
Centered nonagonal number
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A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for n is given by the formula N c =2. Thus, the first few centered nonagonal numbers are 1,10,28,55,91,136,190,253,325,406,496,595,703,820,946, the list above includes the perfect numbers 28 and 496. All even perfect numbers are numbers whose index is an odd Mersenne prime. Since every Mersenne prime greater than 3 is congruent to 1 modulo 3, in 1850, Sir Frederick Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers, which has been neither proven nor disproven
26.
Practical number
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In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. Practical numbers were used by Fibonacci in his Liber Abaci in connection with the problem of representing rational numbers as Egyptian fractions, Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators. The name practical number is due to Srinivasan and he noted that the subdivision of money, weights and measures involved numbers like 4,12,16,20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10. He rediscovered the number theoretical property of such numbers and was the first to attempt a classification of numbers that was completed by Stewart. This characterization makes it possible to determine whether a number is practical by examining its prime factorization, every even perfect number and every power of two is also a practical number. Practical numbers have also shown to be analogous with prime numbers in many of their properties. If the ordered set of all divisors of the number n is d 1, d 2. D j with d 1 =1 and d j = n, in other words the ordered sequence of all divisors d 1 < d 2 <. < d j of a number has to be a complete sub-sequence. This partial characterization was extended and completed by Stewart and Sierpiński who showed that it is straightforward to determine whether a number is practical from its prime factorization, a positive integer greater than one with prime factorization n = p 1 α1. P k α k is if and only if each of its prime factors p i is small enough for p i −1 to have a representation as a sum of smaller divisors. The condition stated above is necessary and sufficient for a number to be practical, in the other direction, the condition is sufficient, as can be shown by induction. Since q ≤ σ and n / p k α k can be shown by induction to be practical, we can find a representation of q as a sum of divisors of n / p k α k. The divisors representing r, together with p k α k times each of the divisors representing q, the only odd practical number is 1, because if n >2 is an odd number, then 2 cannot be expressed as the sum of distinct divisors of n. More strongly, Srinivasan observes that other than 1 and 2, the product of two practical numbers is also a practical number. More strongly the least common multiple of any two numbers is also a practical number. Equivalently, the set of all numbers is closed under multiplication. From the above characterization by Stewart and Sierpiński it can be seen that if n is a practical number, in the set of all practical numbers there is a primitive set of practical numbers
27.
Harmonic divisor number
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In mathematics, a harmonic divisor number, or Ore number, is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are 1,6,28,140,270,496,672,1638,2970,6200,8128,8190, for example, the harmonic divisor number 6 has the four divisors 1,2,3, and 6. Their harmonic mean is an integer,411 +12 +13 +16 =2, the number 140 has divisors 1,2,4,5,7,10,14,20,28,35,70, and 140. All of the terms in this formula are multiplicative, but not completely multiplicative, therefore, the harmonic mean H is also multiplicative. This means that, for any integer n, the harmonic mean H can be expressed as the product of the harmonic means for the prime powers in the factorization of n. For any integer M, as Ore observed, the product of the mean and arithmetic mean of its divisors equals M itself. Therefore, M is harmonic, with mean of divisors k, if. Ore showed that every number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M, therefore, the average of the divisors is M, where τ denotes the number of divisors of M. For any M, τ is odd if and only if M is a square number, for otherwise each divisor d of M can be paired with a different divisor M/d. But, no perfect number can be a square, this follows from the form of even perfect numbers. Therefore, for a perfect number M, τ is even, Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers. W. H. Mills showed that any odd harmonic divisor number above 1 must have a power factor greater than 107. Cohen & Sorli showed that there are no odd harmonic divisor numbers smaller than 1024, Cohen, Goto, and others starting with Ore himself have performed computer searches listing all small harmonic divisor numbers. From these results, lists are known of all harmonic divisor numbers up to 2×109, an Identity Concerning Averages of Divisors of a Given Integer. Numbers Whose Positive Divisors Have Small Integral Harmonic Mean, Cohen, Graeme L. Sorli, Ronald M. Odd harmonic numbers exceed 1024. On Divisors of Odd Perfect Numbers, on the averages of the divisors of a number
28.
Modular arithmetic
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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7,00 now, then 8 hours later it will be 3,00. Usual addition would suggest that the time should be 7 +8 =15. Likewise, if the clock starts at 12,00 and 21 hours elapse, then the time will be 9,00 the next day, because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below,12 is congruent not only to 12 itself, Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers, addition, subtraction, and multiplication. For a positive n, two integers a and b are said to be congruent modulo n, written, a ≡ b. The number n is called the modulus of the congruence, for example,38 ≡14 because 38 −14 =24, which is a multiple of 12. The same rule holds for negative values, −8 ≡72 ≡ −3 −3 ≡ −8. Equivalently, a ≡ b mod n can also be thought of as asserting that the remainders of the division of both a and b by n are the same, for instance,38 ≡14 because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that 38 −14 =24 is a multiple of 12. A remark on the notation, Because it is common to consider several congruence relations for different moduli at the same time, in spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been if the notation a ≡n b had been used. The properties that make this relation a congruence relation are the following, if a 1 ≡ b 1 and a 2 ≡ b 2, then, a 1 + a 2 ≡ b 1 + b 2 a 1 − a 2 ≡ b 1 − b 2. The above two properties would still hold if the theory were expanded to all real numbers, that is if a1, a2, b1, b2. The next property, however, would fail if these variables were not all integers, the notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is referred to as the modulo operation. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12, as this remainder is 2, we have 14 mod 12 =2
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James Joseph Sylvester
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James Joseph Sylvester FRS was an English mathematician. He made fundamental contributions to theory, invariant theory, number theory, partition theory. He played a role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University. At his death, he was professor at Oxford, Sylvester was born James Joseph in London, England. His father, Abraham Joseph, was a merchant, at the age of 14, Sylvester was a student of Augustus De Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife, subsequently, he attended the Liverpool Royal Institution. Sylvester began his study of mathematics at St Johns College, Cambridge in 1831, for the same reason, he was unable to compete for a Fellowship or obtain a Smiths prize. In 1838 Sylvester became professor of philosophy at University College London. In 1841, he was awarded a BA and an MA by Trinity College, following his early retirement, Sylvester published a book entitled The Laws of Verse in which he attempted to codify a set of laws for prosody in poetry. In 1872, he received his B. A. and M. A. from Cambridge. In 1876 Sylvester again crossed the Atlantic Ocean to become the professor of mathematics at the new Johns Hopkins University in Baltimore. His salary was $5,000, which he demanded be paid in gold, after negotiation, agreement was reached on a salary that was not paid in gold. In 1878 he founded the American Journal of Mathematics, the only other mathematical journal in the US at that time was the Analyst, which eventually became the Annals of Mathematics. In 1883, he returned to England to take up the Savilian Professor of Geometry at Oxford University and he held this chair until his death, although in 1892 the University appointed a deputy professor to the same chair. Sylvester invented a number of mathematical terms such as matrix, graph. He coined the term totient for Eulers totient function φ and his collected scientific work fills four volumes. In Discrete geometry he is remembered for Sylvesters Problem and a result on the orchard problem, Sylvester House, a portion of an undergraduate dormitory at Johns Hopkins University, is named in his honor. Several professorships there are named in his honor also, the collected mathematical papers of James Joseph Sylvester, I, New York, AMS Chelsea Publishing, ISBN 978-0-8218-3654-5 Sylvester, James Joseph, Baker, Henry Frederick, ed
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Richard K. Guy
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Richard Kenneth Guy is a British mathematician, professor emeritus in the Department of Mathematics at the University of Calgary. He is known for his work in theory, geometry, recreational mathematics, combinatorics. He is best known for co-authorship of Winning Ways for your Mathematical Plays and he has also published over 300 papers. For this paper he received the MAA Lester R. Ford Award, Guy was born 30 Sept 1916 in Nuneaton, Warwickshire, England, to Adeline Augusta Tanner and William Alexander Charles Guy. Both of his parents were teachers, rising to the rank of headmistress and headmaster and he attended Warwick School for Boys, the third oldest school in Britain, but was not enthusiastic about most of the curriculum. He was good at sports, however, and excelled in mathematics, at the age of 17 he read Dicksons History of the Theory of Numbers. He said it was better than the works of Shakespeare. By then he had developed a passion for mountain climbing. In 1935 Guy entered Gonville and Caius College, at the University of Cambridge as a result of winning several scholarships, to win the most important of these he had to travel to Cambridge and write exams for two days. His interest in games began while at Cambridge where he became a composer of chess problems. In 1938, he graduated with an honours degree, he himself thinks that his failure to get a first may have been related to his obsession with chess. Although his parents advised against it, Guy decided to become a teacher. He met his future wife Nancy Louise Thirian through her brother Michael who was a fellow scholarship winner at Gonville and he and Louise shared loves of mountains and dancing. He wooed her through correspondence, and they married in December 1940, in November 1942, Guy received an emergency commission in the Meteorological Branch of the Royal Air Force, with the rank of flight lieutenant. He was posted to Reykjavik, and later to Bermuda, as a meteorologist and he tried to get permission for Louise to join him but was refused. While in Iceland, he did some glacier travel, skiing and mountain climbing, marking the beginning of another love affair. When Guy returned to England after the war, he went back to teaching, this time at Stockport Grammar School, in 1947 the family moved to London, where he got a job teaching math at Goldsmiths College. In 1951 he moved to Singapore, where he taught at the University of Malaya for the next decade and he then spent a few years at the Indian Institute of Technology in Delhi, India
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Multiplicative inverse
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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity,1. The multiplicative inverse of a fraction a/b is b/a, for the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, the reciprocal function, the function f that maps x to 1/x, is one of the simplest examples of a function which is its own inverse. In the phrase multiplicative inverse, the qualifier multiplicative is often omitted, multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba, then inverse typically implies that an element is both a left and right inverse. The notation f −1 is sometimes used for the inverse function of the function f. For example, the multiplicative inverse 1/ = −1 is the cosecant of x, only for linear maps are they strongly related. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, in the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, the property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no other than 1 and −1 has an integer reciprocal. In modular arithmetic, the multiplicative inverse of a is also defined. This multiplicative inverse exists if and only if a and n are coprime, for example, the inverse of 3 modulo 11 is 4 because 4 ·3 ≡1. The extended Euclidean algorithm may be used to compute it, the sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i. e. nonzero elements x, y such that xy =0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring, the linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case. A ring in which every element has a multiplicative inverse is a division ring. As mentioned above, the reciprocal of every complex number z = a + bi is complex. In particular, if ||z||=1, then 1 / z = z ¯, consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property
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Polite number
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In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. Polite numbers have also called staircase numbers because the Young diagrams representing graphically the partitions of a polite number into consecutive integers resemble staircases. If all numbers in the sum are strictly greater than one, the impolite numbers are exactly the powers of two. It follows from the Lambek–Moser theorem that the nth polite number is ƒ, the politeness of a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For every x, the politeness of x equals the number of odd divisors of x that are greater than one. The politeness of the numbers 1,2,3. is 0,0,1,0,1,1,1,0,2,1,1,1,1,1,3,0,1,2,1,1,3. For instance 90 has politeness 5 because 90 =2 ×32 ×51, the powers of 3 and 5 are respectively 2 and 1, to see the connection between odd divisors and polite representations, suppose a number x has the odd divisor y >1. Then y consecutive integers centered on x/y have x as their sum, some of the terms in this sum may be zero or negative. However, if a term is zero it can be omitted and any negative terms may be used to cancel positive ones, for instance, the polite number x =14 has a single nontrivial odd divisor,7. It is therefore the sum of 7 consecutive numbers centered at 14/7 =2,14 = + + +2 + + +. The first term, −1, cancels a later +1, conversely, every polite representation of x can be formed from this construction. After this extension, again, x/y is the middle term, more generally, the same idea gives a two-to-one correspondence between, on the one hand, representations as a sum of consecutive integers and on the other hand odd divisors. If a polite representation starts with 1, the number so represented is a triangular number T n = n 2 =1 +2 + ⋯ + n. Otherwise, it is the difference of two numbers, i + + + ⋯ + j = T j − T i −1. In the latter case, it is called a trapezoidal number and that is, a trapezoidal number is a polite number that has a polite representation in which all terms are strictly greater than one. Thus, polite non-trapezoidal numbers must have the form of a power of two multiplied by a prime number, for instance, the perfect number 28 =23 −1 and the number 136 =24 −1 are both polite triangular numbers that are not trapezoidal. It is believed there are finitely many Fermat primes, but infinitely many Mersenne primes. Polite Numbers, NRICH, University of Cambridge, December 2002 An Introduction to Runsums, is there any pattern to the set of trapezoidal numbers
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Little-o notation
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Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, in computer science, big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows. Big O notation characterizes functions according to their rates, different functions with the same growth rate may be represented using the same O notation. The letter O is used because the rate of a function is also referred to as order of the function. A description of a function in terms of big O notation usually only provides a bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, Big O notation is also used in many other fields to provide similar estimates. Let f and g be two functions defined on some subset of the real numbers. That is, f = O if and only if there exists a real number M. In many contexts, the assumption that we are interested in the rate as the variable x goes to infinity is left unstated. If f is a product of several factors, any constants can be omitted, for example, let f = 6x4 − 2x3 +5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms, 6x4, −2x3, and 5, of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the rule, 6x4 is a product of 6. Omitting this factor results in the simplified form x4, thus, we say that f is a big-oh of. Mathematically, we can write f = O, one may confirm this calculation using the formal definition, let f = 6x4 − 2x3 +5 and g = x4. Applying the formal definition from above, the statement that f = O is equivalent to its expansion, | f | ≤ M | x 4 | for some choice of x0 and M. To prove this, let x0 =1 and M =13, Big O notation has two main areas of application. In mathematics, it is used to describe how closely a finite series approximates a given function. In computer science, it is useful in the analysis of algorithms, in both applications, the function g appearing within the O is typically chosen to be as simple as possible, omitting constant factors and lower order terms
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Square-free integer
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In mathematics, a square-free, or quadratfrei integer, is an integer which is divisible by no other perfect square than 1. For example,10 is square-free but 18 is not, as 18 is divisible by 9 =32. The smallest positive square-free numbers are 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,39. The radical of an integer is its largest square-free factor, an integer is square-free if and only if it is equal to its radical. Any arbitrary positive integer n can be represented in a way as the product of a powerful number and a square-free integer. The square-free factor is the largest square-free divisor k of n that is coprime with n/k, a positive integer n is square-free if and only if in the prime factorization of n, no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor p of n, also n is square-free if and only if in every factorization n = ab, the factors a and b are coprime. An immediate result of this definition is that all numbers are square-free. A positive integer n is square-free if and only if all abelian groups of n are isomorphic. This follows from the classification of finitely generated abelian groups, a integer n is square-free if and only if the factor ring Z / nZ is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if, for every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice and it is a Boolean algebra if and only if n is square-free. A positive integer n is square-free if and only if μ ≠0, a positive integer n is squarefree if and only if ∑ d 2 ∣ n μ =1. This results from the properties of Möbius function, and the fact that this sum is equal to ∑ d ∣ m μ, where m is the largest divisor of n such that m2 divides n. The Dirichlet generating function for the numbers is ζ ζ = ∑ n =1 ∞ | μ | n s where ζ is the Riemann zeta function. This is easily seen from the Euler product ζ ζ = ∏ p = ∏ p, let Q denote the number of square-free integers between 1 and x. For large n, 3/4 of the positive integers less than n are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Under the Riemann hypothesis, the term can be further reduced to yield Q = x ζ + O =6 x π2 + O
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Deficient number
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In number theory, a deficient or deficient number is a number n for which the sum of divisors σ<2n, or, equivalently, the sum of proper divisors s<n. The value 2n − σ is called the numbers deficiency, as an example, consider the number 21. Its proper divisors are 1,3 and 7, and their sum is 11, because 11 is less than 21, the number 21 is deficient. Its deficiency is 2 ×21 −32 =10, since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. An infinite number of even and odd deficient numbers exist. All odd numbers with one or two prime factors are deficient. All proper divisors of deficient or perfect numbers are deficient, there exists at least one deficient number in the interval for all sufficiently large n. Closely related to deficient numbers are perfect numbers with σ = 2n, the natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica. Almost perfect number Amicable number Sociable number Sándor, József, Mitrinović, Dragoslav S. Crstici, Borislav, the Prime Glossary, Deficient number Weisstein, Eric W. Deficient Number
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Abundant number
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In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number and its proper divisors are 1,2,3,4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance, the number 12 has an abundance of 4, for example. A number n for which the sum of divisors σ>2n, or, equivalently, the sum of proper divisors s>n. The first 28 abundant numbers are,12,18,20,24,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,100,102,104,108,112,114,120, …. For example, the divisors of 24 are 1,2,3,4,6,8. Because 36 is more than 24, the number 24 is abundant and its abundance is 36 −24 =12. The smallest odd abundant number is 945, the smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5,7,11,13,17,19,23, and 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If A represents the smallest abundant number not divisible by the first k primes then for all ϵ >0 we have,2 − ϵ < ln A <2 + ϵ for sufficiently large k, infinitely many even and odd abundant numbers exist. The set of abundant numbers has a natural density, marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480. Every multiple of a number is abundant. For example, every multiple of 6 is abundant because the divisors include 1, n/2, n/3, every multiple of an abundant number is abundant. For example, every multiple of 20 is abundant because n/2 + n/4 + n/5 + n/10 + n/20 = n + n/10, every integer greater than 20161 can be written as the sum of two abundant numbers. An abundant number which is not a number is called a weird number. An abundant number with abundance 1 is called a quasiperfect number, numbers whose sum of proper factors equals the number itself are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers. The abundancy index of n is the ratio σ/n, distinct numbers n1, n2. with the same abundancy index are called friendly numbers. The sequence of least numbers n such that σ > kn, in which a2 =12 corresponds to the first abundant number, if p = is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant