1.
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 By the early century, it had been superseded by number theory.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
2.
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
3.
Ulam spiral
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It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers. Ulam and Gardner emphasized the appearance in the spiral of prominent diagonal, horizontal. Nevertheless, the Ulam spiral is connected with major unsolved problems in number such as Landaus problems. Like Ulam, Klauber noted the connection with prime-generating polynomials, such as Eulers, the number spiral is constructed by writing the positive integers in a spiral arrangement on a square lattice, as shown. In the figure, primes appear to concentrate along certain diagonal lines, in the 200×200 Ulam spiral shown above, diagonal lines are clearly visible, confirming that the pattern continues. Horizontal and vertical lines with a density of primes, while less prominent, are also evident. Starting with 41 at the center gives an impressive example, with a diagonal containing an unbroken string of 40 primes. According to Gardner, Ulam discovered the spiral in 1963 while doodling during the presentation of a long and these hand calculations amounted to a few hundred points. Shortly afterwards, Ulam, with collaborators Myron Stein and Mark Wells, the group also computed the density of primes among numbers up to 10,000,000 along some of the prime-rich lines as well as along some of the prime-poor lines. Images of the spiral up to 65,000 points were displayed on an attached to the machine. The Ulam spiral was described in Martin Gardners March 1964 Mathematical Games column in Scientific American, some of the photographs of Stein, Ulam, and Wells were reproduced in the column. In an addendum to the Scientific American column, Gardner mentioned the earlier paper of Klauber. Klauber describes his construction as follows, The integers are arranged in order with 1 at the apex, the second line containing numbers 2 to 4, the third 5 to 9. Diagonal, horizontal, and vertical lines in the number spiral correspond to polynomials of the form f =4 n 2 + b n + c where b and c are integer constants. When b is even, the lines are diagonal, and either all numbers are odd, or all are even and it is therefore no surprise that all primes other than 2 line in alternate diagonals of the Ulam spiral. To understand why some odd diagonals have a concentration of primes than others. In their 1923 paper on the Goldbach Conjecture, Hardy and Littlewood stated a series of conjectures, one of which, if true, would explain some of the striking features of the Ulam spiral. This conjecture, which Hardy and Littlewood called Conjecture F, is a case of the Bateman–Horn conjecture
4.
Mathematical Association of America
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The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. The MAA was founded in 1915 and is headquartered at 1529 18th Street, Northwest in the Dupont Circle neighborhood of Washington, the organization publishes mathematics journals and books, including the American Mathematical Monthly, the most widely read mathematics journal in the world according to records on JSTOR. The MAA sponsors the annual summer MathFest and cosponsors with the American Mathematical Society the Joint Mathematics Meeting, on occasion the Society for Industrial and Applied Mathematics joins in these meetings. Twenty-nine regional sections also hold regular meetings, the association publishes multiple journals, The American Mathematical Monthly is expository, aimed at a broad audience from undergraduate students to research mathematicians. Mathematics Magazine is expository, aimed at teachers of undergraduate mathematics, the College Mathematics Journal is expository, aimed at teachers of undergraduate mathematics, especially at the freshman-sophomore level. Math Horizons is expository, aimed at undergraduate students, MAA FOCUS is the association member newsletter. The Association publishes an online resource, Mathematical Sciences Digital Library, the service launched in 2001 with the online-only Journal of Online Mathematics and its Applications and a set of classroom tools, Digital Classroom Resources. These were followed in 2004 by Convergence, a history magazine, and in 2005 by MAA Reviews, an online book review service, and Classroom Capsules and Notes. Ultimately, six high school students are chosen to represent the U. S. at the International Mathematics Olympiad. Allendoerfer Award, Trevor Evans Award, Lester R. Ford Award, George Pólya Award, Merten M. Hasse Award, Henry L. Alder Award, a detailed history of the first fifty years of the MAA appears in May. A report on activities prior to World War II appears in Bennet, further details of its history can be found in Case. In addition numerous regional sections of the MAA have published accounts of their local history, the MAA has for a long time followed a strict policy of inclusiveness and non-discrimination. In previous periods it was subject to the problems of discrimination that were widespread across the United States. M. Holloway came to the meeting and were able to attend the scientific sessions, however, the organizer for the closing banquet refused to honor the reservations of these four mathematicians. Lorch and his colleagues wrote to the bodies of the AMS. Bylaws were not changed, but non-discriminatory policies were established and have been observed since then. The Associations first woman president was Dorothy Lewis Bernstein, the Carriage House that belonged to the residents at 1529 18th Street, N. W. dates to around 1900. It is older than the 5-story townhouse where the MAA Headquarters is currently located, charles Evans Hughes occupied the house while he was Secretary of State and a Supreme Court Justice
5.
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
6.
Yuri Matiyasevich
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Yuri Vladimirovich Matiyasevich, is a Russian mathematician and computer scientist. He is best known for his solution of Hilberts tenth problem. As a winner of IMO Yuri Matiyasevich was accepted without exams to LSU, in 1966, he presented a talk at International Congress of Mathematicians held in Moscow. He was an undergraduate student at that time. In 1969–1970, he pursued Ph. D. studies at Leningrad Department of Steklov Institute of Mathematics under supervision of Sergey Maslov, in 1970, he received his Ph. D. degree at LOMI. In 1970–1974, he was a researcher at LOMI, in 1972, he obtained a second doctoral degree. In 1974–1980, he was a researcher at LOMI. Since 1980, Yuri Matiyasevich has been the head of Laboratory of mathematical logic at LOMI, since 1995, he has been a professor of Saint-Petersburg State University, initially at the chair of software engineering, later at the chair of algebra and number theory. In 1997, he was elected as a member of Russian Academy of Sciences. Since 1998, Yuri Matiyasevich has been a vice-president of St. Petersburg Mathematical Society, since 2002, he has been a head of St. Petersburg City Mathematical Olympiad. Since 2003, Matiyasevich has been a co-director of annual German–Russian student school JASS, in 2008, he was elected as a full member of Russian Academy of Sciences. 1964, Gold medal at the International Mathematical Olympiad held in Moscow,1970, Young mathematician prize of the Leningrad Mathematical Society. 1980, Markov Prize of Academy of Sciences of the USSR,1998, He received Humboldt Research Award to Outstanding Scholars. 2003, Honorary Degree, Université Pierre et Marie Curie,2007, Member of the Bayern Academy of Sciences. A polynomial related to the colorings of a triangulation of a sphere was named after Matiyasevich, see The Matiyasevich polynomial, four colour theorem, notable students include, Eldar Musayev, Maxim Vsemirnov, Alexei Pastor, Dmitri Karpov. Yuri Matiyasevich Hilberts 10th Problem, Foreword by Martin Davis and Hilary Putnam, real-time recognition of the inclusion relation. Reduction of an arbitrary Diophantine equation to one in 13 unknowns, decision Problems for Semi-Thue Systems with a Few Rules. Yuri Matiyasevich, Proof Procedures as Bases for Metamathematical Proofs in Discrete Mathematics, Yuri Matiyasevich, Elimination of bounded universal quantifiers standing in front of a quantifier-free arithmetical formula, Personal Journal of Yuri Matiyasevich
7.
American Mathematical Monthly
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The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by the Mathematical Association of America, the American Mathematical Monthly is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content, in this the American Mathematical Monthly fulfills a different role from that of typical mathematical research journals. The American Mathematical Monthly is the most widely read journal in the world according to records on JSTOR. Since 1997, the journal has been available online at the Mathematical Association of Americas website, the MAA gives the Lester R. Ford Awards annually to authors of articles of expository excellence published in the American Mathematical Monthly
8.
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
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.
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