|
Zsigmondy's Theorem
In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a>b>0 are coprime integers, then for any integer n \ge 1, there is a prime number ''p'' (called a ''primitive prime divisor'') that divides a^n-b^n and does not divide a^k-b^k for any positive integer k1 and n is not equal to 6, then 2^n-1 has a prime divisor not dividing any 2^k-1 with k History The theorem was discovered by Zsigmondy working in from 1894 until 1925.Generalizations Let be a sequence of nonzero integers. The Zs ...[...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] |
Fibonacci Sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci ... [...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] |
Elliptic Divisibility Sequence
In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan WardMorgan Ward, Memoir on elliptic divisibility sequences, ''Amer. J. Math.'' 70 (1948), 31–74. in the 1940s. They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including logic and cryptography. Definition A (nondegenerate) ''elliptic divisibility sequence'' (EDS) is a sequence of integers defined recursively by four initial values , , , , with ≠ 0 and with subsequent values det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisibility Sequence
In mathematics, a divisibility sequence is an integer sequence (a_n) indexed by positive integers ''n'' such that :\textm\mid n\texta_m\mid a_n for all ''m'', ''n''. That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined. A strong divisibility sequence is an integer sequence (a_n) such that for all positive integers ''m'', ''n'', :\gcd(a_m,a_n) = a_. Every strong divisibility sequence is a divisibility sequence: \gcd(m,n) = m if and only if m\mid n. Therefore by the strong divisibility property, \gcd(a_m,a_n) = a_m and therefore a_m\mid a_n. Examples * Any constant sequence is a strong divisibility sequence. * Every sequence of the form a_n = kn, for some nonzero integer ''k'', is a divisibility sequence. * The numbers of the form 2^n-1 (Mersenne numbers) form a strong divisibi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lehmer Number
In mathematics, a Lehmer sequence is a generalization of a Lucas sequence. Algebraic relations If ''a'' and ''b'' are complex numbers with :a + b = \sqrt :ab = Q under the following conditions: * ''Q'' and ''R'' are relatively prime nonzero integers * a/b is not a root of unity. Then, the corresponding Lehmer numbers are: :U_n(\sqrt,Q) = \frac for ''n'' odd, and :U_n(\sqrt,Q) = \frac for ''n'' even. Their companion numbers are: :V_n(\sqrt,Q) = \frac for ''n'' odd and :V_n(\sqrt,Q) = a^n+b^n for ''n'' even. Recurrence Lehmer numbers form a linear recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ... with :U_n = (R-2Q)U_-Q^2U_ = (a^2+b^2)U_-a^2b^2U_ with initial values U_0=0,\, U_1=1,\, U_2=1,\, U_3=R-Q=a^2+ab+b^2. Similarly the companion sequence sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U_n(P, Q) and V_n(P, Q). More generally, Lucas sequences U_n(P, Q) and V_n(P, Q) represent sequences of polynomials in P and Q with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers . Lucas sequences are named after the French mathematician Édouard Lucas. Recurrence relations Given two integer parameters P and Q, the Lucas sequences of the first kind U_n(P,Q) and of the second kind V_n(P,Q) are defined by the recurrence relations: :\begin U_0(P,Q)&=0, \\ U_1(P,Q)&=1, \\ U_n(P,Q)&=P\cdot U_(P,Q)-Q\cdot U_(P,Q) \mbox ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pell Sequence
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , and , so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82. Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + . As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carmichael's Theorem
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind ''U''''n''(''P'', ''Q'') with relatively prime parameters ''P'', ''Q'' and positive discriminant, an element ''U''''n'' with ''n'' ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = ''U''12(1, −1) = 144 and its equivalent ''U''12(−1, −1) = −144. In particular, for ''n'' greater than 12, the ''n''th Fibonacci number F(''n'') has at least one prime divisor that does not divide any earlier Fibonacci number. Carmichael (1913, Theorem 21) proved this theorem. Recently, Yabuta (2001) gave a simple proof. Statement Given two relatively prime integers ''P'' and ''Q'', such that D=P^2-4Q>0 and , let be the Lucas sequence of the first kind defined by :\begin U_0(P,Q)&=0, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Zsigmondy
Karl Zsigmondy () (27 March 1867 – 14 October 1925) was an Austrian mathematician of Hungarian ethnicity. He was a son of Adolf Zsigmondy from Pozsony, Kingdom of Hungary (now Bratislava, Slovakia) and his mother was Irma von Szakmáry of Martonvásár, Kingdom of Hungary. He studied (1886–1890) and worked (1894–1925) at the University of Vienna. After his PhD, in 1890, he studied at the University of Berlin, University of Göttingen and at the Sorbonne in Paris, but came back to Vienna in 1894. He discovered Zsigmondy's theorem in 1892. He was the brother of the mountain climber Emil Zsigmondy and the Nobel Laureate chemist Richard Adolf Zsigmondy Richard Adolf Zsigmondy ( hu, Zsigmondy Richárd Adolf; 1 April 1865 – 23 September 1929) was an Austrian-born chemist. He was known for his research in colloids, for which he was awarded the Nobel Prize in chemistry in 1925, as well as for c .... References * * External links * {{DEFAULTSORT:Zsigmondy, Karl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vienna
en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST = CEST , utc_offset_DST = +2 , blank_name = Vehicle registration , blank_info = W , blank1_name = GDP , blank1_info = € 96.5 billion (2020) , blank2_name = GDP per capita , blank2_info = € 50,400 (2020) , blank_name_sec1 = HDI (2019) , blank_info_sec1 = 0.947 · 1st of 9 , blank3_name = Seats in the Federal Council , blank3_info = , blank_name_sec2 = GeoTLD , blank_info_sec2 = .wien , website = , footnotes = , image_blank_emblem = Wien logo.svg , blank_emblem_size = Vienna ( ; german: Wien ; ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications On Pure And Applied Mathematics
''Communications on Pure and Applied Mathematics'' is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited by the institute, typically in the fields of applied mathematics, mathematical analysis, or mathematical physics. The journal was established in 1948 as the ''Communications on Applied Mathematics'', obtaining its current title the next year. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 3.219. References External links * Mathematics journals Monthly journals Wiley (publisher) academic journals Publications established in 1948 English-lang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
