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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] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from 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 can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...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] |
Heuristic Argument
A heuristic argument is an argument that reasons from the value of a method or principle that has been shown experimentally (especially through trial-and-error) to be useful or convincing in learning, discovery and problem-solving, but whose line of reasoning involves key oversimplifications that make it not entirely rigorous. A widely used and important example of a heuristic argument is Occam's Razor. It is a speculative, non-rigorous argument that relies on analogy or intuition, and that allows one to achieve a result or an approximation that is to be checked later with more rigor. Otherwise, the results are generally to be doubted. It is used as a hypothesis or a conjecture in an investigation, though it can also be used as a mnemonic as well. See also *Empirical relationship *Heuristic *Probabilistic method *Rule of thumb In English language, English, the phrase ''rule of thumb'' refers to an approximate method for doing something, based on practical experience rather than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist. Notation In formulas, a limit of a function is usually written as : \lim_ f(x) = L, and is read as "the limit of of as approaches equals ". This means that the value of the function can be made arbitrarily close to , by choosing sufficiently close to . Alternatively, the fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dickson's Conjecture
In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by that for a finite set of linear forms , , ..., with , there are infinitely many positive integers for which they are all prime, unless there is a congruence condition preventing this . The case ''k'' = 1 is Dirichlet's theorem. Two other special cases are well-known conjectures: there are infinitely many twin primes (''n'' and 2 + ''n'' are primes), and there are infinitely many Sophie Germain primes (''n'' and 1 + 2''n'' are primes). Generalized Dickson's conjecture Given ''n'' polynomials with positive degrees and integer coefficients (''n'' can be any natural number) that each satisfy all three conditions in the Bunyakovsky conjecture, and for any prime ''p'' there is an integer ''x'' such that the values of all ''n'' polynomials at ''x'' are not divisible by ''p'', then there are infinitely many positive integers ''x'' such that all values of these ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sexy Prime
In number theory, sexy primes are prime numbers that differ from each other by . For example, the numbers and are a pair of sexy primes, because both are prime and 11 - 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six: . If or (where is the lower prime) is also prime, then the sexy prime is part of a prime triplet. In August 2014, the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin A cousin is a relative who is the child of a parent's sibling; this is more specifically referred to as a first cousin. A parent of a first cousin is an aunt or uncle. More generally, in the kinship system used in the English-speaking world, ... or sexy primes. The sexy primes (sequences and in OEIS) below 500 are: :(5,11), (7,13) ... [...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] |
Twin Prime Conjecture
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 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polymath Projects
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 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alphonse De Polignac
Alphonse de Polignac (1826–1863) was a French mathematician and aristocrat. He is known for Polignac's Conjecture. Biography His father, Jules de Polignac (1780-1847) was prime minister of Charles X until the Bourbon dynasty was overthrown in the July Revolution of 1830. Alphonse was born in London during his father's time as ambassador to the United Kingdom. In 1849 he was admitted to Polytechnique and went onto serve in the Crimean War as an artillery officer, achieving the rank of Captain. He was also a historian, a poet, a musician, and authored a translation of the play Faust by Goethe. His work in mathematics mainly focused on Number Theory and he specifically worked with prime numbers. Polignac's Conjecture In his first year at Polytechnique Polignac formulated his eponymous conjecture, which states that: For every positive integer ''k'', there are infinitely many prime gaps of size 2''k''. Other work in Mathematics Polignac also conjectured Romanov's The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Maynard (mathematician)
James Alexander Maynard (born 10 June 1987) is an English mathematician working in analytic number theory and in particular the theory of prime numbers. In 2017, he was appointed Research Professor at Oxford. Maynard is a fellow of St John's College, Oxford. He was awarded the Fields Medal in 2022 and the New Horizons in Mathematics Prize in 2023. Education Maynard attended King Edward VI Grammar School, Chelmsford in Chelmsford, England. After completing his bachelor's and master's degrees at Queens' College, Cambridge, in 2009, Maynard obtained his D.Phil. from Balliol College, Oxford, in 2013 under the supervision of Roger Heath-Brown. He then became a Fellow by Examination at Magdalen College, Oxford. Career For the 2013–2014 year, Maynard was a CRM-ISM postdoctoral researcher at the University of Montreal. In November 2013, Maynard gave a different proof of Yitang Zhang's theorem that there are bounded gaps between primes, and resolved a longstanding conjectu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
