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Lenstra–Pomerance–Wagstaff Conjecture
In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one. Original Mersenne conjecture The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his ''Cogitata Physico-Mathematica'' (1644; see e.g. Dickson 1919) that the numbers 2^n - 1 were prime for ''n'' = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers ''n'' ≤ 257. Due to the size of these numbers, Mersenne did not and could not test all of them, nor could his peers in the 17th century. It was eventually determined, after three centuries and the availability of new techniques such as the Lucas–Lehmer test, that Mersenne's conjecture contained five errors, namely two are composite (those corresponding to the primes ''n'' = 67, 257) and three omitted primes (those corresponding to the primes ''n'' = 61, 89, 107). The correct list is: ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hendrik Lenstra
Hendrik Willem Lenstra Jr. (born 16 April 1949, Zaandam) is a Dutch mathematician. Biography Lenstra received his doctorate from the University of Amsterdam in 1977 and became a professor there in 1978. In 1987 he was appointed to the faculty of the University of California, Berkeley; starting in 1998, he divided his time between Berkeley and the University of Leiden, until 2003, when he retired from Berkeley to take a full-time position at Leiden. Three of his brothers, Arjen Lenstra, Andries Lenstra, and Jan Karel Lenstra, are also mathematicians. Jan Karel Lenstra is the former director of the Netherlands Centrum Wiskunde & Informatica (CWI). Hendrik Lenstra was the Chairman of the Program Committee of the International Congress of Mathematicians in 2010. Scientific contributions Lenstra has worked principally in computational number theory. He is well known for: * Co-discovering of the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (in 1982); * Developing a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to . Then the theorem states that \pi(x)\sim\frac. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. Definition Formally, given functions and , we define a binary relation f(x) \sim g(x) \qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures About Prime Numbers
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of The Theory Of Numbers
''History of the Theory of Numbers'' is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, with little further discussion. The central topic of quadratic reciprocity and higher reciprocity laws is barely mentioned; this was apparently going to be the topic of a fourth volume that was never written . Volumes * Volume 1 - Divisibility and Primality - 486 pages * Volume 2 - Diophantine Analysis In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ... - 803 pages * Volume 3 - Quadratic and Higher Forms - 313 pages References * * * * * * * * * * * * External links History of the Theory of Numbers - Volume 1at the Internet Archive. History of the Theory of Numbers - Volu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for 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 mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne's Laws
Mersenne's laws are laws describing the frequency of oscillation of a stretched string or monochord, useful in musical tuning and musical instrument construction. Overview The equation was first proposed by French mathematician and music theorist Marin Mersenne in his 1636 work ''Harmonie universelle''.Mersenne, Marin (1636)''Harmonie universelle'' Cited in, '' Wolfram.com''. Mersenne's laws govern the construction and operation of string instruments, such as pianos and harps, which must accommodate the total tension force required to keep the strings at the proper pitch. Lower strings are thicker, thus having a greater mass per length. They typically have lower tension. Guitars are a familiar exception to this: string tensions are similar, for playability, so lower string pitch is largely achieved with increased mass per length. Higher-pitched strings typically are thinner, have higher tension, and may be shorter. "This result does not differ substantially from Galileo's, yet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan's Mersenne Conjecture
In mathematics, a double Mersenne number is a Mersenne number of the form :M_ = 2^-1 where ''p'' is prime. Examples The first four terms of the sequence of double Mersenne numbers areChris Caldwell''Mersenne Primes: History, Theorems and Lists''at the Prime Pages. : :M_ = M_3 = 7 :M_ = M_7 = 127 :M_ = M_ = 2147483647 :M_ = M_ = 170141183460469231731687303715884105727 Double Mersenne primes A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number ''M''''p'' can be prime only if ''p'' is prime, (see Mersenne prime for a proof), a double Mersenne number M_ can be prime only if ''M''''p'' is itself a Mersenne prime. For the first values of ''p'' for which ''M''''p'' is prime, M_ is known to be prime for ''p'' = 2, 3, 5, 7 while explicit factors of M_ have been found for ''p'' = 13, 17, 19, and 31. Thus, the smallest candidate for the next double Mersenne prime is M_, or 22305843009213693951 − 1. Being approximately 1.695, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Primality Test
In computational number theory, the Lucas test is a primality test for a natural number ''n''; it requires that the prime factors of ''n'' − 1 be already known. It is the basis of the Pratt certificate that gives a concise verification that ''n'' is prime. Concepts Let ''n'' be a positive integer. If there exists an integer ''a'', 1 < ''a'' < ''n'', such that : and for every prime factor ''q'' of ''n'' − 1 : then ''n'' is prime. If no such number ''a'' exists, then ''n'' is either 1, 2, or . The reason for the correctness of this claim is as follows: if the first equivalence holds for ''a'', we can deduce that ''a'' and ''n'' are |
Gillies' Conjecture
In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good and Daniel Shanks. The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted (but also open) Lenstra–Pomerance–Wagstaff conjecture. The conjecture :\textA < B < \sqrt\textB/A\textM_p \rightarrow \infty\textM : :: He noted that his conjecture would imply that # The number of Mersenne primes less than is . # The expect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourth Power
In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So: :''n''4 = ''n'' × ''n'' × ''n'' × ''n'' Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares. The sequence of fourth powers of integers (also known as biquadrates or tesseractic numbers) is: :0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, ... . Properties The last digit of a fourth power in decimal can only be 0 (in fact 0000), 1, 5 (in fact 0625), or 6. Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as the sum of at most 16 fourth powers (see Waring's problem). Fermat knew that a fourth power cannot be the sum of two other fourth powers (the ''n'' = 4 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
