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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] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Hardy–Littlewood Conjecture
In number theory, the first Hardy–Littlewood conjecture states the asymptotic formula for the number of prime ''k''-tuples less than a given magnitude by generalizing the prime number theorem. It was first proposed by G. H. Hardy and John Edensor Littlewood in 1923.. Statement Let m_1, m_2, \ldots, m_k be positive even integers such that the numbers of the sequence P = (p, p + m_1, p + m_2, \ldots , p + m_k) do not form a complete residue class with respect to any prime and let \pi_(n) denote the number of primes p less than n st. p + m_1, p + m_2, \ldots , p + m_k are all prime. Then :\pi_P(n)\sim C_P\int_2^n \frac, where :C_P=2^k \prod_\frac is a product over odd primes and w(q; m_1, m_2, \ldots , m_k) denotes the number of distinct residues of 0, m_1, m_2, \ldots , m_k modulo q. The case k=1 and m_1=2 is related to the twin prime conjecture. Specifically if \pi_2(n) denotes the number of twin primes less than ''n'' then :\pi_2(n)\sim C_2 \int_2^n \frac, where :C_2 = 2\pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Zero
In mathematics, more specifically in the field of analytic number theory, a Landau–Siegel zero or simply Siegel zero, also known as exceptional zeroSee Iwaniec (2006).), named after Edmund Landau and Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions associated to quadratic number fields. Roughly speaking, these are possible zeros very near (in a quantifiable sense) to s=1. (The pronunciation of "Siegel" begins with a Z sound.) Motivation and definition The way in which Siegel zeros appear in the theory of Dirichlet L-functions is as potential exceptions to the classical zero-free regions, which can only occur when the L-function is associated to a real Dirichlet character. Real primitive Dirichlet characters For an integer , a Dirichlet character modulo is an arithmetic function \chi\colon \mathbb\to\mathbb satisfying the following properties: * ( Completely multiplicative) \chi(mn) = \chi( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldbach’s Conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than but remains unproven despite considerable effort. History Origins On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (""), in which Goldbach had remarked that the first of those two conjectures would follow from the statement This is in fact equivalent to his seco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliott–Halberstam Conjecture
In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated a specific version of the conjecture in 1968. One version of the conjecture is as follows, and stating it requires some notation. Let \pi(x), the prime-counting function, denote the number of primes less than or equal to x. If q is a positive integer and a is coprime to q, we let \pi(x;q,a) denote the number of primes less than or equal to x which are equal to a modulo q. Dirichlet's theorem on primes in arithmetic progressions then tells us that : \pi(x;q,a) \sim \frac\ \ (x\rightarrow\infty) where \varphi is Euler's totient function. If we then define the error function : E(x;q) = \max_ \left, \pi(x;q,a) - \frac\ where the max is taken over all a coprime to q, then the Elliott–Halberstam conjecture is the assertion that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polymath Project
The Polymath Project is a collaboration among mathematicians to solve important and difficult mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution. The project began in January 2009 on Timothy Gowers's blog when he posted a problem and asked his readers to post partial ideas and partial progress toward a solution. This experiment resulted in a new answer to a difficult problem, and since then the Polymath Project has grown to describe a particular crowdsourcing process of using an online collaboration to solve any math problem. Origin In January 2009, Gowers chose to start a social experiment on his blog by choosing an important unsolved mathematical problem and issuing an invitation for other people to help solve it collaboratively in the comments section of his blog. Along with the math problem itself, Gowers asked a question which was included in the title of his blog post, "is massively collaborative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nature (journal)
''Nature'' is a British weekly scientific journal founded and based in London, England. As a multidisciplinary publication, ''Nature'' features Peer review, peer-reviewed research from a variety of academic disciplines, mainly in science and technology. It has core editorial offices across the United States, continental Europe, and Asia under the international scientific publishing company Springer Nature. ''Nature'' was one of the world's most cited scientific journals by the Science Edition of the 2022 ''Journal Citation Reports'' (with an ascribed impact factor of 50.5), making it one of the world's most-read and most prestigious academic journals. , it claimed an online readership of about three million unique readers per month. Founded in the autumn of 1869, ''Nature'' was first circulated by Norman Lockyer and Alexander MacMillan (publisher), Alexander MacMillan as a public forum for scientific innovations. The mid-20th century facilitated an editorial expansion for the j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Polignac's Conjecture
In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: :For any positive even number ''n'', there are infinitely many prime gaps of size ''n''. In other words: There are infinitely many cases of two consecutive prime numbers with difference ''n''. Although the conjecture has not yet been proven or disproven for any given value of ''n'', in 2013 an important breakthrough was made by Yitang Zhang who proved that there are infinitely many prime gaps of size ''n'' for some value of ''n'' < 70,000,000. Later that year, James Maynard announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600. As of April 14, 2014, one year after Zhang's announcement, according to the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
