1.
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
2.
Prime gap
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A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g is the difference between the -th and the prime numbers, i. e. g n = p n +1 − p n. We have g1 =1, g2 = g3 =2, the sequence of prime gaps has been extensively studied, however many questions and conjectures remain unanswered. By the definition of gn every prime can be written as p n +1 =2 + ∑ i =1 n g i. The first, smallest, and only odd prime gap is 1 between the only prime number,2, and the first odd prime,3. All other prime gaps are even, there is only one pair of gaps between three consecutive odd natural numbers for which all are prime. These gaps are g2 and g3 between the primes 3,5, and 7, for any prime number P, we write P# for P primorial, that is, the product of all prime numbers up to and including P. Therefore, there exist gaps between primes that are large, i. e. for any prime number P. Another way to see that arbitrarily large prime gaps must exist is the fact that the density of primes approaches zero, in fact, by this theorem, P# is very roughly a number the size of exp, and near exp the average distance between consecutive primes is P. In reality, prime gaps of P numbers can occur at much smaller than P#. Although the average gap between primes increases as the logarithm of the integer, the ratio of the prime gap to the integers involved decreases. This is a consequence of the prime number theorem, see below, on the other hand, the ratio of the gap to the number of digits of the integers involved does increase without bound. This is a consequence of a result by Westzynthius, see below, in the opposite direction, the twin prime conjecture asserts that gn =2 for infinitely many integers n. As of March 2017 the largest known prime gap with identified probable prime gap ends has length 5103138, with 216849-digit probable primes found by Robert W. Smith. The largest known prime gap with identified proven primes as gap ends has length 1113106, with 18662-digit primes found by P. Cami, M. Jansen and we say that gn is a maximal gap, if gm < gn for all m < n. As of August 2016 the largest known maximal gap has length 1476 and it is the 75th maximal gap, and it occurs after the prime 1425172824437699411. Other record maximal gap terms can be found at A002386, usually the ratio of gn / ln is called the merit of the gap gn. In 1931, E. Westzynthius proved that maximal prime gaps grow more than logarithmically and that is, lim sup n → ∞ g n log p n = ∞
3.
Yitang Zhang
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Yitang Tom Zhang is a Chinese-born American mathematician working in the area of number theory. This work led to a 2014 MacArthur award and his appointment as a professor, Zhang was born in Shanghai and lived there until he was 13 years old. At around the age of nine, he found a proof of the Pythagorean theorem and he first learned about Fermat’s last theorem and the Goldbach conjecture when he was 10. During the Cultural Revolution, he and his mother were sent to the countryside to work in the fields and he worked as a laborer for 10 years and was unable to attend high school. After the Cultural Revolution ended, Zhang entered Peking University in 1978 as an undergraduate student and he became a graduate student of Professor Pan Chengbiao, a number theorist at Peking University, and obtained his M. Sc. degree in mathematics in 1984. Zhang arrived at Purdue in January 1985, studied there for seven years, Zhangs Ph. D. work was on the Jacobian conjecture. After graduation, Zhang had a hard time finding an academic position, in a 2013 interview with Nautilus magazine, Zhang said he did not get a job after graduation. During that period it was difficult to find a job in academics and that was a job market problem. Also, my advisor did not write me letters of recommendation, the reason behind this is that Zhangs research pointed out the mistakes made by his advisor Tzuong-Tsieng Mohs previous work. Moh was very unhappy with this and refused to write the job recommendation letter for Zhang, Zhang made this claim again in George Csicsery’s documentary film Counting From Infinity while discussing his difficulties at Purdue and in the years that followed. Tzuong-Tsieng Moh, his Ph. D. advisor at Purdue, after some years, Zhang managed to find a position as a lecturer at the University of New Hampshire, where he was hired by Kenneth Appel in 1999. Prior to getting back to academia, he worked for years as an accountant. He also worked in a motel in Kentucky and in a Subway sandwich shop, a profile published in the Quanta Magazine reports that Zhang used to live in his car during the initial job-hunting days. He served as lecturer at UNH from 1999 until around January 2014, in Fall 2015, Zhang accepted an offer of full professorship at the University of California, Santa Barbara. On April 17,2013, Zhang announced a proof that there are infinitely many pairs of prime numbers that differ by 70 million or less. This result implies the existence of an infinitely repeatable prime 2-tuple, Zhangs paper was accepted by Annals of Mathematics in early May 2013, his first publication since his last paper in 2001. The proof was refereed by leading experts in analytic number theory, Zhangs result set off a flurry of activity in the field, such as the Polymath8 project. The classical form of the twin prime conjecture is equivalent to P, while these stronger conjectures remain unproven, a result due to James Maynard in November 2013, employing a different technique, showed that P holds for some k ≤600
4.
James Maynard (mathematician)
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James Maynard is a British mathematician best known for his work on prime gaps. This work can be seen as progress on the Hardy–Littlewood m -tuples conjecture as it establishes that a proportion of admissible m -tuples satisfy the prime m -tuples conjecture for every m. Maynards approach yielded the upper bound lim inf n → ∞ ≤600, subsequently, Polymath 8b was created, whose collaborative efforts have reduced the gap size to 252. As of April 14,2014, one year after Zhangs announcement, according to the Polymath project wiki, further, assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that N has been reduced to 12 and 6, respectively. In August 2014, Maynard resolved a conjecture of Erdős on large gaps between primes, and received the largest Erdős prize ever offered. After completing his bachelors and masters degrees at University of Cambridge in 2009, from University of Oxford at Balliol College in 2013 under the supervision of Roger Heath-Brown. For the 2013–2014 year, Maynard was a CRM-ISM postdoctoral researcher at the University of Montreal, in 2014, he was awarded the SASTRA Ramanujan Prize
5.
Terence Tao
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Terence Terry Chi-Shen Tao FAA FRS, is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Tao was a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics. Taos father, Dr. Billy Tao, was a pediatrician who was born in Shanghai, Taos mother, Grace, is from Hong Kong, she received a first-class honours degree in physics and mathematics at the University of Hong Kong. She was a school teacher of mathematics and physics in Hong Kong. Billy and Grace met as students at the University of Hong Kong and they then emigrated from Hong Kong to Australia. Tao has two living in Australia, both of whom represented Australia at the International Mathematical Olympiad. Nigel Tao was part of the team at Google Australia that created Google Wave and he now works on the Go programming language. Trevor Tao is an International Master in Chess and he has a double degree in mathematics and music and is an autistic savant. Taos wife, Laura, is an engineer at NASAs Jet Propulsion Laboratory and they live with their son and daughter in Los Angeles, California. Tao exhibited extraordinary mathematical abilities from an age, attending university level mathematics courses at the age of 9. In 1986,1987, and 1988, Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten, winning a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiads history, winning the gold medal shortly after his thirteenth birthday, at age 14, Tao attended the Research Science Institute. When he was 15 he published his first assistant paper and he received his bachelors and masters degrees at the age of 16 from Flinders University under Garth Gaudry. In 1992 he won a Fulbright Scholarship to undertake study in the United States. From 1992 to 1996, Tao was a student at Princeton University under the direction of Elias Stein. He joined the faculty of the University of California, Los Angeles in 1996, when he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution. Within the field of mathematics, Tao is known for his collaboration with Ben J. Green of Oxford University, known for his collaborative mindset, by 2006 Tao had worked with over 30 others in his discoveries, reaching 68 co-authors by October 2015
6.
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
7.
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
8.
Prime number theorem
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In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 using ideas introduced by Bernhard Riemann, the first such distribution found is π ~ N/log, where π is the prime-counting function and log is the natural logarithm of N. This means that for large enough N, the probability that an integer not greater than N is prime is very close to 1 / log. Consequently, an integer with at most 2n digits is about half as likely to be prime as a random integer with at most n digits. For example, among the integers of at most 1000 digits, about one in 2300 is prime, whereas among positive integers of at most 2000 digits. In other words, the gap between consecutive prime numbers among the first N integers is roughly log. Let π be the function that gives the number of primes less than or equal to x. For example, π =4 because there are four prime numbers less than or equal to 10, using asymptotic notation this result can be restated as π ∼ x log x. This notation does not say anything about the limit of the difference of the two functions as x increases without bound, instead, the theorem states that x / log x approximates π in the sense that the relative error of this approximation approaches 0 as x increases without bound. For example, the 7017200000000000000♠2×1017th prime number is 7018851267738604819♠8512677386048191063, and log rounds to 7018796741875229174♠7967418752291744388, a relative error of about 6. 4%. The prime number theorem is equivalent to lim x → ∞ ϑ x = lim x → ∞ ψ x =1, where ϑ and ψ are the first. Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in 1797 or 1798 that π is approximated by the function a /, where A and B are unspecified constants. In the second edition of his book on number theory he made a more precise conjecture. Carl Friedrich Gauss considered the question at age 15 or 16 in the year 1792 or 1793. In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, in two papers from 1848 and 1850, the Russian mathematician Pafnuty Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. He was able to prove unconditionally that this ratio is bounded above, an important paper concerning the distribution of prime numbers was Riemanns 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject. In particular, it is in paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π originates
9.
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
10.
Brun's theorem
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In number theory, Bruns theorem states that the sum of the reciprocals of the twin primes converges to a finite value known as Bruns constant, usually denoted by B2. Bruns theorem was proved by Viggo Brun in 1919, and it has importance in the introduction of sieve methods. The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes, let π2 denote the number of primes p ≤ x for which p +2 is also prime. Then, for x ≥3, we have π2 = O and that is, twin primes are less frequent than prime numbers by nearly a logarithmic factor. It follows from this bound that the sum of the reciprocals of the primes converges, or stated in other words. In explicit terms the sum ∑ p, p +2 ∈ P = + + + ⋯ either has finitely many terms or has many terms but is convergent. The fact that the sum of the reciprocals of the prime numbers diverges implies that there are many prime numbers. Because the sum of the reciprocals of the twin primes instead converges, Bruns constant could be an irrational number only if there are infinitely many twin primes. By calculating the twin primes up to 1014, Thomas R. Nicely heuristically estimated Bruns constant to be 1.902160578, Nicely has extended his computation to 1. 6×1015 as of 18 January 2010 but this is not the largest computation of its type. In 2002 Pascal Sebah and Patrick Demichel used all twin primes up to 1016 to give the estimate and it is based on extrapolation from the sum 1.830484424658. for the twin primes below 1016. Dominic Klyve showed conditionally that B2 <2.1754, there is also a Bruns constant for prime quadruplets. A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 and this constant should not be confused with the Bruns constant for cousin primes, as prime pairs of the form, which is also written as B4. Wolf derived an estimate for the Brun-type sums Bn of 4/n, let C2 =0.6601 … be the twin prime constant. Then it is conjectured that π2 ∼2 C2 x 2, in particular, π2 < x 2 for every ε >0 and all sufficiently large x. Many special cases of the above have been proved, most recently, Jie Wu proved that for sufficiently large x, π2 <4.5 x 2 where 4.5 corresponds to ε ≈3.18 in the above. The digits of Bruns constant were used in a bid of $1,902,160,540 in the Nortel patent auction, the bid was posted by Google and was one of three Google bids based on mathematical constants. Divergence of the sum of the reciprocals of the primes Meissel–Mertens constant Weisstein, Sebah, Pascal and Xavier Gourdon, Introduction to twin primes and Bruns constant computation,2002
11.
Mathematical constant
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A mathematical constant is a special number, usually a real number, that is significantly interesting in some way. Constants arise in areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory. The more popular constants have been studied throughout the ages and computed to many decimal places, all mathematical constants are definable numbers and usually are also computable numbers. These are constants which one is likely to encounter during pre-college education in many countries, however, its ubiquity is not limited to pure mathematics. It appears in many formulas in physics, and several physical constants are most naturally defined with π or its reciprocal factored out and it is debatable, however, if such appearances are fundamental in any sense. For example, the textbook nonrelativistic ground state wave function of the atom is ψ =11 /2 e − r / a 0. This formula contains a π, but it is unclear if that is fundamental in a physical sense, furthermore, this formula gives only an approximate description of physical reality, as it omits spin, relativity, and the quantal nature of the electromagnetic field itself. The numeric value of π is approximately 3.1415926535, memorizing increasingly precise digits of π is a world record pursuit. The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth, suppose a slot machine with a one in n probability of winning is played n times. Then, for large n the probability that nothing will be won is approximately 1/e, another application of e, discovered in part by Jacob Bernoulli along with French mathematician Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem. Here n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into labelled boxes, the butler does not know the name of the guests, and so must put them into boxes selected at random. The problem of de Montmort is, what is the probability that none of the hats gets put into the right box, the answer is p n =1 −11. + ⋯ + n 1 n. and as n tends to infinity, the numeric value of e is approximately 2.7182818284. The square root of 2, often known as root 2, radical 2, or Pythagorass constant, and written as √2, is the algebraic number that. It is more called the principal square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational and its numerical value truncated to 65 decimal places is,1.41421356237309504880168872420969807856967187537694807317667973799. The quick approximation 99/70 for the root of two is frequently used
12.
Helmut Maier
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Helmut Maier is a German mathematician and professor at the University of Ulm, Germany. He is known for his contributions in number theory and mathematical analysis. He has also done important work in exponential sums and trigonometric sums over special sets of integers and his research has been groundbreaking and deeply influential. Helmut Maier graduated with a Diploma in Mathematics from the University of Ulm in 1976 and he received his Ph. D. from the University of Minnesota in 1981, under the supervision of J. Ian Richards. Maiers Ph. D. thesis was an extension of his paper H. Maier, in this paper Maier applied for the first time what is now known as Maiers matrix method. This method later on led him and other mathematicians to the discovery of unexpected irregularities in the distribution of prime numbers, there have been various other applications of Maiers Matrix Method, such as on irreducible polynomials and on strings of consecutive primes in the same residue class. After postdoctoral positions at the University of Michigan and the Institute for Advanced Study, Princeton, while in Georgia he proved that the usual formulation of the Cramér model for the distribution of prime numbers is wrong. This was an unexpected result. Jointly with Carl Pomerance he studied the values of Eulers φ-function, during the same period Maier investigated as well the size of the coefficients of cyclotomic polynomials and later collaborated with Sergei Konyagin and E. Wirsing on this topic. He also collaborated with Hugh Lowell Montgomery on the size of the sum of the Möbius function under the assumption of the Riemann Hypothesis, Maier and Gérald Tenenbaum in joint work investigated the sequence of divisors of integers, solving the famous propinquity problem of Paul Erdős. Since 1993 Maier is a Professor at the University of Ulm, rassias have worked on problems related to the Estermann zeta function and the Nyman-Beurling criterion on the Riemann Hypothesis involving the distribution of certain cotangent sums. Other collaborators of Helmut Maier include Paul Erdős, C. Feiler, John Friedlander, Andrew Granville, D. Haase, A. J. Hildebrand, Michel Laurent Lapidus, J. W. Neuberger, A. Sankaranarayanan, A. Sárközy, Wolfgang P. Schleich, Maiers matrix method Maiers theorem Maiers webpage
13.
Daniel Goldston
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Daniel Alan Goldston is an American mathematician who specializes in number theory. He is currently a professor of mathematics at San Jose State University and he has an Erdős number of 2. P n +1 − p n < c log p n and this result was originally reported in 2003 by Goldston and Yıldırım but was later retracted. Then Pintz joined the team and they completed the proof in 2005, in fact, if they assume the Elliott–Halberstam conjecture, then they can also show that primes within 16 of each other occur infinitely often, which is related to the twin prime conjecture. Landaus problems Yitang Zhang James Maynard Dan Goldstons Homepage
14.
G. H. Hardy
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Godfrey Harold G. H. Hardy FRS was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, Hardy is known for the Hardy–Weinberg principle, a principle of population genetics. In addition to his research, Hardy is remembered for his 1940 essay on the aesthetics of mathematics and he was the mentor of the Indian mathematician Srinivasa Ramanujan. Starting in 1914, Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan, Hardy almost immediately recognised Ramanujans extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was and he called their collaboration the one romantic incident in my life. G. H. Hardy was born on 7 February 1877, in Cranleigh, Surrey, England and his father was Bursar and Art Master at Cranleigh School, his mother had been a senior mistress at Lincoln Training College for teachers. Hardys own natural affinity for mathematics was perceptible at an early age, when just two years old, he wrote numbers up to millions, and when taken to church he amused himself by factorising the numbers of the hymns. After schooling at Cranleigh, Hardy was awarded a scholarship to Winchester College for his mathematical work, in 1896 he entered Trinity College, Cambridge. After only two years of preparation under his coach, Robert Alfred Herman, Hardy was fourth in the Mathematics Tripos examination. Years later, he sought to abolish the Tripos system, as he felt that it was becoming more an end in itself than a means to an end, while at university, Hardy joined the Cambridge Apostles, an elite, intellectual secret society. In 1900 he passed part II of the tripos and was awarded a fellowship, in 1903 he earned his M. A. which was the highest academic degree at English universities at that time. From 1906 onward he held the position of a lecturer where teaching six hours per week left him time for research, in 1919 he left Cambridge to take the Savilian Chair of Geometry at Oxford in the aftermath of the Bertrand Russell affair during World War I. Hardy spent the academic year 1928–1929 at Princeton in an exchange with Oswald Veblen. Hardy gave the Josiah Willards Gibbs lecture for 1928, Hardy left Oxford and returned to Cambridge in 1931, where he was Sadleirian Professor until 1942. The Indian Clerk is a novel by David Leavitt based on Hardys life at Cambridge, including his discovery of, Hardy is credited with reforming British mathematics by bringing rigour into it, which was previously a characteristic of French, Swiss and German mathematics. British mathematicians had remained largely in the tradition of applied mathematics, from 1911 he collaborated with John Edensor Littlewood, in extensive work in mathematical analysis and analytic number theory. This led to progress on the Warings problem, as part of the Hardy–Littlewood circle method. In prime number theory, they proved results and some notable conditional results and this was a major factor in the development of number theory as a system of conjectures, examples are the first and second Hardy–Littlewood conjectures
15.
Limit of a function
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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below, informally, a function f assigns an output f to every input x. We say the function has a limit L at an input p, more specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a distance apart. The notion of a limit has many applications in modern calculus, in particular, the many definitions of continuity employ the limit, roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative, in the calculus of one variable, however, his work was not known during his lifetime. Weierstrass first introduced the definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0, the modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908. Imagine a person walking over a landscape represented by the graph of y = f and her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y and she is walking towards the horizontal position given by x = p. As she gets closer and closer to it, she notices that her altitude approaches L, if asked about the altitude of x = p, she would then answer L. What, then, does it mean to say that her altitude approaches L. It means that her altitude gets nearer and nearer to L except for a small error in accuracy. For example, suppose we set a particular goal for our traveler. She reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, the accuracy goal is then changed, can she get within one vertical meter. If she is anywhere within seven meters of p, then her altitude always remains within one meter from the target L. This explicit statement is quite close to the definition of the limit of a function with values in a topological space. To say that lim x → p f = L, means that ƒ can be made as close as desired to L by making x close enough, the following definitions are the generally accepted ones for the limit of a function in various contexts. Suppose f, R → R is defined on the real line, the value of the limit does not depend on the value of f, nor even that p be in the domain of f
16.
Integration by parts
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It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be derived in one line simply by integrating the product rule of differentiation, more general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts, the theorem can be derived as follows. Suppose u and v are two differentiable functions. The product rule states, d d x = v d d x + u d d x and it is not actually necessary for u and v to be continuously differentiable. Integration by parts works if u is continuous and the function designated v is Lebesgue integrable. This is only if we choose v = − exp . One can also come up with similar examples in which u and v are not continuously differentiable. This visualisation also explains why integration by parts may help find the integral of an inverse function f−1 when the integral of the f is known. Indeed, the x and y are inverses, and the integral ∫x dy may be calculated as above from knowing the integral ∫y dx. The following form is useful in illustrating the best strategy to take, as a simple example, consider, ∫ ln x 2 d x. Since the derivative of ln is 1/x, one makes part u, since the antiderivative of 1/x2 is -1/x, the formula now yields, ∫ ln x 2 d x = − ln x − ∫ d x. The antiderivative of −1/x2 can be found with the rule and is 1/x. Alternatively, one may choose u and v such that the product u simplifies due to cancellation, for example, suppose one wishes to integrate, ∫ sec 2 ⋅ ln d x. The integrand simplifies to 1, so the antiderivative is x, finding a simplifying combination frequently involves experimentation. Some other special techniques are demonstrated in the examples below, exponentials and trigonometric functions An example commonly used to examine the workings of integration by parts is I = ∫ e x cos d x. Here, integration by parts is performed twice, then, ∫ e x sin d x = e x sin − ∫ e x cos d x. Putting these together, ∫ e x cos d x = e x cos + e x sin − ∫ e x cos d x, the same integral shows up on both sides of this equation
17.
Density function
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In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. The probability density function is everywhere, and its integral over the entire space is equal to one. The terms probability distribution function and probability function have also sometimes used to denote the probability density function. However, this use is not standard among probabilists and statisticians, further confusion of terminology exists because density function has also been used for what is here called the probability mass function. In general though, the PMF is used in the context of random variables. Suppose a species of bacteria typically lives 4 to 6 hours, what is the probability that a bacterium lives exactly 5 hours. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.0000000000, instead we might ask, What is the probability that the bacterium dies between 5 hours and 5.01 hours. Lets say the answer is 0.02, next, What is the probability that the bacterium dies between 5 hours and 5.001 hours. The answer is probably around 0.002, since this is 1/10th of the previous interval, the probability that the bacterium dies between 5 hours and 5.0001 hours is probably about 0.0002, and so on. In these three examples, the ratio / is approximately constant, and equal to 2 per hour, for example, there is 0.02 probability of dying in the 0. 01-hour interval between 5 and 5.01 hours, and =2 hour−1. This quantity 2 hour−1 is called the probability density for dying at around 5 hours, therefore, in response to the question What is the probability that the bacterium dies at 5 hours. A literally correct but unhelpful answer is 0, but an answer can be written as dt. This is the probability that the bacterium dies within a window of time around 5 hours. For example, the probability that it lives longer than 5 hours, there is a probability density function f with f =2 hour−1. The integral of f over any window of time is the probability that the dies in that window. A probability density function is most commonly associated with absolutely continuous univariate distributions, a random variable X has density fX, where fX is a non-negative Lebesgue-integrable function, if, Pr = ∫ a b f X d x. That is, f is any function with the property that. In the continuous univariate case above, the measure is the Lebesgue measure
18.
Distributed computing
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Distributed computing is a field of computer science that studies distributed systems. A distributed system is a model in which components located on networked computers communicate and coordinate their actions by passing messages, the components interact with each other in order to achieve a common goal. Three significant characteristics of distributed systems are, concurrency of components, lack of a global clock, examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications. A computer program that runs in a system is called a distributed program. There are many alternatives for the message passing mechanism, including pure HTTP, RPC-like connectors, Distributed computing also refers to the use of distributed systems to solve computational problems. In distributed computing, a problem is divided into many tasks, each of which is solved by one or more computers, which communicate with each other by message passing. The terms are used in a much wider sense, even referring to autonomous processes that run on the same physical computer. The entities communicate with each other by message passing, a distributed system may have a common goal, such as solving a large computational problem, the user then perceives the collection of autonomous processors as a unit. Other typical properties of distributed systems include the following, The system has to tolerate failures in individual computers. The structure of the system is not known in advance, the system may consist of different kinds of computers and network links, each computer has only a limited, incomplete view of the system. Each computer may know one part of the input. Distributed systems are groups of networked computers, which have the goal for their work. The terms concurrent computing, parallel computing, and distributed computing have a lot of overlap, the same system may be characterized both as parallel and distributed, the processors in a typical distributed system run concurrently in parallel. Parallel computing may be seen as a tightly coupled form of distributed computing. In distributed computing, each processor has its own private memory, Information is exchanged by passing messages between the processors. The figure on the right illustrates the difference between distributed and parallel systems, figure shows a parallel system in which each processor has a direct access to a shared memory. The situation is complicated by the traditional uses of the terms parallel and distributed algorithm that do not quite match the above definitions of parallel. The use of concurrent processes that communicate by message-passing has its roots in operating system architectures studied in the 1960s, the first widespread distributed systems were local-area networks such as Ethernet, which was invented in the 1970s
19.
PrimeGrid
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PrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform, PrimeGrid started in June 2005 under the name Message@home and tried to decipher text fragments hashed with MD5. Message@home was a test to port the BOINC scheduler to Perl to obtain greater portability, after a while the project attempted the RSA factoring challenge trying to factor RSA-640. After RSA-640 was factored by a team in November 2005. With the chance to succeed too small, it discarded the RSA challenges, was renamed to PrimeGrid, at 210,000,000,000 the primegen subproject was stopped. In June 2006, dialog started with Riesel Sieve to bring their project to the BOINC community, PrimeGrid provided PerlBOINC support and Riesel Sieve was successful in implementing their sieve as well as a prime finding application. With collaboration from Riesel Sieve, PrimeGrid was able to implement the LLR application in partnership with another prime finding project, in November 2006, the TPS LLR application was officially released at PrimeGrid. Less than two months later, January 2007, the twin was found by the original manual project. PrimeGrid and TPS then advanced their search for even larger twin primes, the summer of 2007 was very active as the Cullen and Woodall prime searches were launched. In the Fall, more prime searches were added through partnerships with the Prime Sierpinski Problem, additionally, two sieves were added, the Prime Sierpinski Problem combined sieve which includes supporting the Seventeen or Bust sieve, and the combined Cullen/Woodall sieve. In the Fall of 2007, PrimeGrid migrated its systems from PerlBOINC to standard BOINC software, since September 2008, PrimeGrid is also running a Proth prime sieving subproject. In January 2010 the subproject Seventeen or Bust was added, the calculations for the Riesel problem followed in March 2010. In addition, PrimeGrid is helping test for a record Sophie Germain prime. As of March 2016, PrimeGrid is working on or has worked on the projects,321 Prime Search is a continuation of Paul Underwoods 321 Search which looked for primes of the form 3 · 2n −1. PrimeGrid added the +1 form and continues the search up to n = 25M, the search was successful in April 2010 with the finding of the first known AP26,43142746595714191 +23681770 · 23# · n is prime for n =0. 23# = 2·3·5·7·11·13·17·19·23 =223092870, or 23 primorial, is the product of all primes up to 23, PrimeGrid is also running a search for Cullen prime numbers, yielding the two largest known Cullen primes. The first one being the 14th largest known prime at the time of discovery, as of 9 March 2014 PrimeGrid has eliminated 14 values of k from the Riesel problem and is continuing the search to eliminate the 50 remaining numbers. Primegrid then worked with the Twin Prime Search to search for a twin prime at approximately 58700 digits
20.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
21.
2 (number)
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2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set
22.
37 (number)
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37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37
23.
53 (number)
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53 is the natural number following 52 and preceding 54. Fifty-three is the 16th prime number and it is also an Eisenstein prime, and a Sophie Germain prime. The sum of the first 53 primes is 5830, which is divisible by 53,53 written in hexadecimal is 35, that is, the same characters used in the decimal representation, but reversed. Four multiples of 53 share this property,371 =17316,5141 =141516,99481 =1849916, and 8520280 =82025816,53 cannot be expressed as the sum of any integer and its base-10 digits, making 53 a self number. 53 is the smallest prime number that does not divide the order of any sporadic group, the duration of Saros series 53 was 1514.5 years, and it contained 85 solar eclipses. The Saros number of the lunar eclipse series began on June 5,993 BC. The duration of Saros series 53 was 1280.1 years, fictional 53rd Precinct in the Bronx was found in the TV comedy Car 54, Where Are You. UDP and TCP port number for the Domain Name System protocol, 53-TET is a musical temperament that has a fifth that is closer to pure than our current system. 53 More Things To Do In Zero Gravity is a mentioned in The Hitchhikers Guide to the Galaxy. 53 a number used on the hand of the tulip in Infinity Train
24.
83 (number)
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83 is the natural number following 82 and preceding 84. 83 is, the sum of three consecutive primes, the sum of five consecutive primes. The 23rd prime number, following 79 and preceding 89, an Eisenstein prime with no imaginary part and real part of the form 3n −1. The duration of Saros series 83 was 1262.1 years, the Saros number of the lunar eclipse series which began on -197 August 22 and ended on 1318 February. The duration of Saros series 83 was 1514.5 years, when someone reaches 83 they may celebrate a second bar mitzvah M83 is the debut album of the French electronic music group M8383 is a song written by John Mayer in the Room for Squares album. As an example, the television station CIVIC-TV managed by the James Woods character Max Renn in the 1983 film Videodrome was on Channel 83. Eighty-three is also, The year AD83,83 BC, or 1983 The TI-83 series and this symbology is also known to be used by many non-racist Christians and non-denominational Churches. An emoticon based on,3 with wide-open eyes
25.
89 (number)
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89 is the natural number following 88 and preceding 90. 89 is, the 24th prime number, following 83 and preceding 97, the smallest Sophie Germain prime to start a Cunningham chain of the first kind of six terms. An Eisenstein prime with no part and real part of the form 3n −1. A Fibonacci number and thus a Fibonacci prime as well, the first few digits of its reciprocal coincide with the Fibonacci sequence due to the identity 189 = ∑ n =1 ∞ F ×10 − =0.011235955 …. A Markov number, appearing in solutions to the Markov Diophantine equation with other odd-indexed Fibonacci numbers, M89 is the 10th Mersenne prime. Although 89 is not a Lychrel number in base 10, it is unusual that it takes 24 iterations of the reverse, among the known non-Lychrel numbers in the first 10000 integers, no other number requires that many or more iterations. The palindrome reached is also unusually large, eighty-nine is, The atomic number of actinium. Messier object M89, a magnitude 11.5 elliptical galaxy in the constellation Virgo, the New General Catalogue object NGC89, a magnitude 13.5 peculiar spiral galaxy in the constellation Phoenix and a member of Roberts Quartet. The Oklahoma Redhawks, an American minor league team, were formerly known as the Oklahoma 89ers. The number alludes to the Land Run of 1889, when the Unassigned Lands of Oklahoma were opened to white settlement, the teams home of Oklahoma City was founded during this event. In Rugby, an 89 or eight-nine move is a following a scrum, in which the number 8 catches the ball. The Elite 89 Award is presented by the U. S. NCAA to the participant in each of the NCAAs 89 championship finals with the highest grade point average. The jersey number 89 has been retired by three National Football League teams in honor of past playing greats, The Baltimore Colts, for Hall of Famer Gino Marchetti, the franchise continues to honor the number in its current identity as the Indianapolis Colts. The Boston Patriots, for Bob Dee, the franchise, now the New England Patriots, continues to honor the number. The Chicago Bears, for Mike Ditka, eighty-nine is also, The designation of Interstate 89, a freeway that runs from New Hampshire to Vermont The designation of U. S. The number of units of each colour in the board game Blokus The number of the French department Yonne Information Is Beautiful cites eighty-nine as one of the words censored on the Chinese internet
26.
Almost all
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In mathematics, the phrase almost all has a number of specialised uses which extend its intuitive meaning. Almost all is used synonymously with all but finitely many or all but a countable set. A simple example is that almost all numbers are odd. Perversely, if we allow almost all to all but a countable set, then it follows that almost all prime numbers are even. When speaking about the reals, sometimes it means all reals, in this sense almost all reals are not a member of the Cantor set even though the Cantor set is uncountable. More generally, almost all is used in the sense of almost everywhere in measure theory. Thus, almost all positive integers are composite, however there are still a number of primes. Generic property Sufficiently large Weisstein, Eric W