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Irregular 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. Kummer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series e = \sum\limits_^ \frac = 1 + \frac + \frac + \frac + \cdots. It is also the unique positive number such that the graph of the function has a slope of 1 at . The (natural) exponential function is the unique function that equals its own derivative and satisfies the equation ; hence one can also define as . The natural logarithm, or logarithm to base , is the inverse function to the natural exponential function. The natural logarithm of a number can be defined directly as the area under the curve between and , in which case is the value of for which this area equals one (see image). There are various other characteriz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harry Vandiver
Harry Schultz Vandiver (21 October 1882 – 9 January 1973) was an American mathematician, known for work in number theory. He was born in Philadelphia, Pennsylvania to John Lyon and Ida Frances (Everett) Vandiver. He did not complete a formal education, choosing instead to leave school at an early age to work for his father's firm, although he did attend some graduate classes at the University of Pennsylvania in 1904–5. From 1917 to 1919 he was a member of the United States Naval Reserve, and in 1919 became an instructor of mathematics at Cornell University, where he taught for five years before becoming an associate professor of pure mathematics at the University of Texas in 1924. He was made a full professor the following year, and named distinguished professor of applied mathematics and astronomy in 1947. He remained at Texas until his retirement in 1966. Vandiver won the Frank Nelson Cole Prize of the American Mathematical Society for his paper on Fermat's Last Theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Number
In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion :\frac = \frac = \sum_^\infty \frac \cdot t^n, where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely: :E_n=2^nE_n(\tfrac 12). The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements. Examples The odd-indexed Euler numbers are all zero. The even-indexed ones have alternating signs. Some values are: : Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive . This article adheres to the convention adopted above. Explicit formulas In terms of Stirling numbers of the second kind Foll ... [...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. In base 10, 1572 is 24649, and 1582 is 24964, which uses the same digits. Numbers having this property are listed in . The previous entry is 13, and the next entry after 157 is 913. The simplest right angle triangle with rational sides that has area 157 has the longest side with a denominator of 45 digits. In the military * was a United States Coast Guard cutter built in 1926 * was a United State ... [...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" (shortened as "iff") is a biconditional logical connective between statements, 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'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...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. Initially state-owned, since 1991 it has been a private company. The company is a member of International Associa ... [...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, o ... [...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 was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to 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 influential in in ... [...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 definer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
