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Prime Constellation
In number theory, a prime -tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a -tuple , the positions where the -tuple matches a pattern in the prime numbers are given by the set of integers such that all of the values are prime. Typically the first value in the -tuple is 0 and the rest are distinct positive even numbers. Named patterns Several of the shortest ''k''-tuples are known by other common names: OEIS sequence covers 7-tuples (''prime septuplets'') and contains an overview of related sequences, e.g. the three sequences corresponding to the three admissible 8-tuples (''prime octuplets''), and the union of all 8-tuples. The first term in these sequences corresponds to the first prime in the smallest prime constellation shown below. Admissibility In order for a -tuple to have infinitely many positions at which all of its values are prime, there cannot exist a prime such that the tuple includes every d ... [...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] |
Admissibility , in ...
Admissibility may refer to: Law * Admissible evidence, evidence which may be introduced in a court of law *Admissibility (ECHR), whether a case will be considered in the European Convention on Human Rights system Mathematics and logic * Admissible decision rule, in statistical decision theory, a rule which is never dominated * Admissible rule, in logic, a type of rule of inference * Admissible heuristic, in computer science, is a heuristic which is no more than the lowest-cost path to the goal * Admissible prime k-tuple, in number theory regarding possible constellations of prime numbers * Admissible set, in mathematical logic, a transitive set satisfying the axioms of Kripke-Platek set theory * Admissible representation In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra. Real or comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skewes' Number
In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which :\pi(x) > \operatorname(x), where is the prime-counting function and is the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing between \pi(x) \operatorname(x) near e^ < 1.397 \times 10^. It is not known whether it is the smallest crossing. Skewes's numbers , who was Skewes's research supervisor, had proved in that there is such a number (and so, a first such number); and in ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skewes's Number
In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which :\pi(x) > \operatorname(x), where is the prime-counting function and is the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing between \pi(x) \operatorname(x) near e^ < 1.397 \times 10^. It is not known whether it is the smallest crossing. Skewes's numbers , who was Skewes's research supervisor, had proved in that there is such a number (and so, a first such number); and in ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310. The first five primorials are: : 2, 6, 30, 210, 2310 . The sequence also includes as empty product. Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Hardy–Littlewood Conjecture
In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the first Hardy–Littlewood conjecture, the second Hardy–Littlewood conjecture was proposed by G. H. Hardy and John Edensor Littlewood in 1923.. Statement The conjecture states that \pi(x+y) \leq \pi(x) + \pi(y) for integers , where denotes the prime-counting function, giving the number of prime numbers up to and including . Connection to the first Hardy–Littlewood conjecture The statement of the second Hardy–Littlewood conjecture is equivalent to the statement that the number of primes from to is always less than or equal to the number of primes from 1 to . This was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime -tuples, and the first violation is expected to likely occur for very large values of . For example, an admissible ''k''-tuple (or prime constellation In number theory, a prime -tuple is a finite collection of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Hardy–Littlewood 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 (41, 43). 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 (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is chairman of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 350,000 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. History Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punched cards. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A008407
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version can be written in two forms: the double-storey a and single-storey ɑ. The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English grammar, " a", and its variant " an", are indefinite articles. History The earliest certain ancestor of "A" is aleph (also written 'aleph), the first letter of the Phoenician alphabet, which consisted entirely of consonants (for that reason, it is also called an abjad to distinguish it fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yitang Zhang
Yitang Zhang (; born February 5, 1955) is a Chinese American mathematician primarily working on number theory and a professor of mathematics at the University of California, Santa Barbara since 2015. Previously working at the University of New Hampshire as a lecturer, Zhang submitted a paper to the ''Annals of Mathematics'' in 2013 which established the first finite bound on the least gap between consecutive primes that is attained infinitely often. This work led to a 2013 Ostrowski Prize, a 2014 Cole Prize, a 2014 Rolf Schock Prize, and a 2014 MacArthur award. Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015. Early life and education Zhang was born in Shanghai, China, with his ancestral home in Pinghu, Zhejiang. He lived in Shanghai with his grandmother until he went to Peking University. At around the age of nine, he found a proof of the Pythagorean theorem. He first learned about Fermat's Last Theorem and the Goldbach conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Constellations
In number theory, a prime -tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a -tuple , the positions where the -tuple matches a pattern in the prime numbers are given by the set of integers such that all of the values are prime. Typically the first value in the -tuple is 0 and the rest are distinct positive even numbers. Named patterns Several of the shortest ''k''-tuples are known by other common names: OEIS sequence covers 7-tuples (''prime septuplets'') and contains an overview of related sequences, e.g. the three sequences corresponding to the three admissible 8-tuples (''prime octuplets''), and the union of all 8-tuples. The first term in these sequences corresponds to the first prime in the smallest prime constellation shown below. Admissibility In order for a -tuple to have infinitely many positions at which all of its values are prime, there cannot exist a prime such that the tuple includes every d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
