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Regular Prime
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers. The first few regular odd primes are: : 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ... . History and motivation In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent ''p'' if ''p'' is regular. This focused attention on the irregular primes. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent ''p'', if is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either or fails to be an irregular pair. ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Number
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
157 (number)
157 (one hundred ndfifty-seven) is the number following 156 and preceding 158. In mathematics 157 is: * the 37th prime number. The next prime is 163 and the previous prime is 151. * a balanced prime, because the arithmetic mean of those primes yields 157 * an emirp * a Chen prime * the largest known prime ''p'' which \frac is also prime. (See ). * the least irregular prime with index 2. * a palindromic number in bases 7 (3137) and 12 (11112). * a repunit in base 12, so it is a unique prime in the same base * a prime whose digits sum to a prime. (see ) * a prime index prime * a super-prime Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. In other words, if prime numbers are matched ... (37 is prime) In base 10, 1572 is 24649, and 1582 is 24964, which uses the same digits. Numbers having this property are listed in . The pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Scientific Publishers PWN
Wydawnictwo Naukowe PWN (''Polish Scientific Publishers PWN''; until 1991 ''Państwowe Wydawnictwo Naukowe'' - ''National Scientific Publishers PWN'', PWN) is a Polish book publisher, founded in 1951, when it split from the Wydawnictwa Szkolne i Pedagogiczne. Adam Bromberg, who was the enterprise's director between 1953 and 1965, made it into communist Poland's largest publishing house. The printing house is best known as a publisher of encyclopedias, dictionaries and university handbooks. It is the leading Polish provider of scientific, educational and professional literature as well as works of reference. It authored the Wielka Encyklopedia Powszechna PWN, by then the largest Polish encyclopedia, as well as its successor, the Wielka Encyklopedia PWN, which was published between 2001 and 2005. There is also an online PWN encyclopedia – Internetowa encyklopedia PWN ''Internetowa encyklopedia PWN'' (Polish language, Polish for ''Internet PWN Encyclopedia'') is a free online ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Carlitz
Leonard Carlitz (December 26, 1907 – September 17, 1999) was an American mathematician. Carlitz supervised 44 doctorates at Duke University and published over 770 papers. Chronology * 1907 Born Philadelphia, PA, USA * 1927 BA, University of Pennsylvania * 1930 PhD, University of Pennsylvania, 1930 under Howard Mitchell, who had studied under Oswald Veblen at Princeton * 1930–31 at Caltech with E. T. Bell * 1931 married Clara Skaler * 1931–32 at Cambridge with G. H. Hardy * 1932 Joined the faculty of Duke University where he served for 45 years * 1938 to 1973 Editorial Board Duke Mathematical Journal (Managing Editor from 1945.) * 1939 Birth of son Michael * 1940 Supervision of his first doctoral student E. F. Canaday, awarded 1940 * 1945 Birth of son Robert * 1964 First James B. Duke Professor in Mathematics * 1977 Supervised his 44th and last doctoral student, Jo Ann Lutz, awarded 1977 * 1977 Retired * 1990 Death of wife Clara, after 59 years of marriage * 1999 Sep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niels Nielsen (mathematician)
Niels Nielsen (2 December 1865, in Ørslev – 16 September 1931, in Copenhagen) was a Danish mathematician who specialised in mathematical analysis. Life and work Nielsen was the son of humble peasants and grew up in the western part of the island of Funen. In 1891 he graduated in mathematics from the University of Copenhagen and in 1895 obtained his doctorate. In 1909 he succeeded Julius Petersen as Professor of Mathematics at the University of Copenhagen. His most original works were on special functions, with an important contribution to the theory of the gamma function. In 1917 he suffered from an illness from which he never fully recovered. From this date onward he became interested in the number theory, Bernoulli numbers, Stirling numbers, and the history of mathematics, writing two books on Danish mathematicians of the time period 1528-1908, and two other books on French mathematicians. Selected publications * ''Om en klasse bestemte integraler og nogle derved define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaj Løchte Jensen
Kaj may refer to: Places * Kaj River, a river of Afghanistan Places in Iran * Kaj, Chaharmahal and Bakhtiari * Kaj, Hamadan * Kaj, Isfahan * Kaj, Qom * Kaj, Razavi Khorasan * Kaj, Sistan and Baluchestan People and characters * Kaj (name) * A fictional frog on the Danish TV series ''Kaj & Andrea'' Other uses * KAJ (group), a musical trio from Finland * ''Kaj'', a conjunction in Esperanto See also * KAJ (other) Kaj may refer to: Places * Kaj River, a river of Afghanistan Places in Iran * Kaj, Chaharmahal and Bakhtiari * Kaj, Hamadan * Kaj, Isfahan * Kaj, Qom * Kaj, Razavi Khorasan * Kaj, Sistan and Baluchestan People and characters * Kaj (name) * A fict ... * * * Kai (other) * Kay (other) {{disambiguation, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number is a mathematical constant approximately equal to 2.71828 that is the base of a logarithm, base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted \gamma. Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. The number is of great importance in mathematics, alongside 0, 1, Pi, , and . All five appear in one formulation of Euler's identity e^+1=0 and play important and recurring roles across mathematics. Like the constant , is Irrational number, irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is Transcendental number, transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
