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Brun's Constant
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods. Asymptotic bounds on twin primes The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let \pi_2(x) denote the number of primes ''p'' ≤ ''x'' for which ''p'' + 2 is also prime (i.e. \pi_2(x) is the number of twin primes with the smaller at most ''x''). Then, we have :\pi_2(x) = O\!\left(\frac \right)\!. That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor. This bound gives the intuition that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form a small set. In explicit terms, the sum :\su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Riemann Hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these ''L''-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case). Global ''L''-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meissel–Mertens Constant
The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as the Mertens constant, Kronecker's constant (after Leopold Kronecker), Hadamard–de la Vallée-Poussin constant (after Jacques Hadamard and Charles Jean de la Vallée-Poussin), or the prime reciprocal constant, is a mathematical constant in number theory, defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm: :M = \lim_ \left( \sum_ \frac - \ln(\ln n) \right)=\gamma + \sum_ \left \ln\! \left( 1 - \frac \right) + \frac \right Here γ is the Euler–Mascheroni constant, which has an analogous definition involving a sum over all integers (not just the primes). The value of ''M'' is approximately :''M'' ≈ 0.2614972128476427837554268386086958590516... . Mertens' second theorem establishes that the limit exists. The fact that there are two logarithms (log of a log) in the limit for the Meissel� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergence Of The Sum Of The Reciprocals Of The Primes
The sum of the reciprocals of all prime numbers diverges; that is: \sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series). There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that \sum_\frac1p \ge \log \log (n+1) - \log\frac6 for all natural numbers . The double natural logarithm () indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant. The harmonic series First, we will describe how Euler originally discovered the result. He was considering the harmonic series \sum_^\infty \frac = 1 + \frac + \frac + \frac + \cdots = \infty He had already used the following " product formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intel
Intel Corporation is an American multinational corporation and technology company headquartered in Santa Clara, California, and Delaware General Corporation Law, incorporated in Delaware. Intel designs, manufactures, and sells computer components such as central processing units (CPUs) and related products for business and consumer markets. It is one of the world's List of largest semiconductor chip manufacturers, largest semiconductor chip manufacturers by revenue, and ranked in the Fortune 500, ''Fortune'' 500 list of the List of largest companies in the United States by revenue, largest United States corporations by revenue for nearly a decade, from 2007 to 2016 Fiscal year, fiscal years, until it was removed from the ranking in 2018. In 2020, it was reinstated and ranked 45th, being the List of Fortune 500 computer software and information companies, 7th-largest technology company in the ranking. It was one of the first companies listed on Nasdaq. Intel supplies List of I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Public Relations
Public relations (PR) is the practice of managing and disseminating information from an individual or an organization (such as a business, government agency, or a nonprofit organization) to the public in order to influence their perception. Public relations and publicity differ in that PR is controlled internally, whereas publicity is not controlled and contributed by external parties. Public relations may include an organization or individual gaining exposure to their audiences using topics of public interest and news items that do not require direct payment. The exposure is mostly media-based, and this differentiates it from advertising as a form of marketing communications. Public relations often aims to create or obtain coverage for clients for free, also known as earned media, rather than paying for marketing or advertising also known as paid media. However, advertising, especially of the type that focuses on distributing information or core PR messages, is also a part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Constant
A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as and pi, occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (). Other constants are notable more for historical reasons than for their mathematical properties. The more popular constants have been studied throughout the ages and computed to many decimal places. All named mathematical constants are Definable real number, definable numbers, and usually are also computable numbers (Chaitin's constant being a significant exception). Basic mathematical constants These a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Google
Google LLC (, ) is an American multinational corporation and technology company focusing on online advertising, search engine technology, cloud computing, computer software, quantum computing, e-commerce, consumer electronics, and artificial intelligence (AI). It has been referred to as "the most powerful company in the world" by the BBC and is one of the world's List of most valuable brands, most valuable brands. Google's parent company, Alphabet Inc., is one of the five Big Tech companies alongside Amazon (company), Amazon, Apple Inc., Apple, Meta Platforms, Meta, and Microsoft. Google was founded on September 4, 1998, by American computer scientists Larry Page and Sergey Brin. Together, they own about 14% of its publicly listed shares and control 56% of its stockholder voting power through super-voting stock. The company went public company, public via an initial public offering (IPO) in 2004. In 2015, Google was reorganized as a wholly owned subsidiary of Alphabet Inc. Go ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nortel
Nortel Networks Corporation (Nortel), formerly Northern Telecom Limited, was a Canadian Multinational corporation, multinational telecommunications and data networking equipment manufacturer headquartered in Ottawa, Ontario. It was founded in Montreal, Quebec in 1895 as the Northern Electric and Manufacturing Company, or simply Northern Electric. Until an antitrust settlement in 1949, Northern Electric was owned mostly by Bell Canada and the Western Electric, Western Electric Company of the Bell System, producing large volumes of telecommunications equipment based on licensed Western Electric designs. At its height, Nortel accounted for more than a third of the total valuation of all companies listed on the Toronto Stock Exchange (TSX), employing 94,500 people worldwide. In 2009, Nortel filed for bankruptcy protection in Canada and the United States, triggering a 79% decline in its corporate stock price. The bankruptcy case was the List of corporate collapses and scandals, larg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 101 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair is not considered to be a pair of twin primes. Since 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cousin Prime
In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. The cousin primes (sequences and in OEIS) below 1000 are: :(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281), (307, 311), (313, 317), (349, 353), (379, 383), (397, 401), (439, 443), (457, 461), (463,467), (487, 491), (499, 503), (613, 617), (643, 647), (673, 677), (739, 743), (757, 761), (769, 773), (823, 827), (853, 857), (859, 863), (877, 881), (883, 887), (907, 911), (937, 941), (967, 971) Properties The only prime belonging to two pairs of cousin primes is 7. One of the numbers will always be divisible by 3, so is the only case where all three are primes. An example of a large proven cousin prime pair is for :p = 4111286921397 \times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
