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Parity Problem (sieve Theory)
In number theory, the parity problem refers to a limitation in sieve theory that prevents sieves from giving good estimates in many kinds of prime-counting problems. The problem was identified and named by Atle Selberg in 1949. Beginning around 1996, John Friedlander and Henryk Iwaniec developed some parity-sensitive sieves that make the parity problem less of an obstacle. Statement Terence Tao gave this "rough" statement of the problem: This problem is significant because it may explain why it is difficult for sieves to "detect primes," in other words to give a non-trivial lower bound for the number of primes with some property. For example, in a sense Chen's theorem is very close to a solution of the twin prime conjecture, since it says that there are infinitely many primes ''p'' such that ''p'' + 2 is either prime or the product of two primes. The parity problem suggests that, because the case of interest has an odd number of prime factors (namely 1), it won't be possible to se ... [...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] |
Selberg Sieve
In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. Description In terms of sieve theory the Selberg sieve is of ''combinatorial type'': that is, derives from a careful use of the inclusion–exclusion principle. Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem. The result gives an ''upper bound'' for the size of the sifted set. Let A be a set of positive integers \le x and let P be a set of primes. Let A_d denote the set of elements of A divisible by d when d is a product of distinct primes from P. Further let A_1 denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are \le z. The object of the sieve is to estimate :S(A,P,z) = \left\vert A \setminus \bigc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A may alternatively be denoted by n(A), , \operatorname(A), or \#A. History A crude sense of cardinality, an awareness that groups of things or events compare with other grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primes In Arithmetic Progression
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a_n = 3 + 4n for 0 \le n \le 2. According to the Green–Tao theorem, there exist arbitrarily long sequences of primes in arithmetic progression. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form an + b, where ''a'' and ''b'' are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For integer ''k'' ≥ 3, an AP-''k'' (also called PAP-''k'') is any sequence of ''k'' primes in arithmetic progression. An AP-''k'' can be written as ''k'' primes of the form ''a''·''n'' + ''b'', for fixed integers ''a'' (called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anatolii Alexeevitch Karatsuba
Anatoly Alexeyevich Karatsuba (his first name often spelled Anatolii) (russian: Анато́лий Алексе́евич Карацу́ба; Grozny, Soviet Union, 31 January 1937 – Moscow, Russia, 28 September 2008) was a Russian people, Russian mathematician working in the field of analytic number theory, p-adic number, ''p''-adic numbers and Dirichlet series. For most of his student and professional life he was associated with the MSU Faculty of Mechanics and Mathematics, Faculty of Mechanics and Mathematics of Moscow State University, defending a Doktor nauk, D.Sc. there entitled "The method of trigonometric sums and intermediate value theorems" in 1966. He later held a position at the Steklov Institute of Mathematics of the Russian Academy of Sciences, Academy of Sciences. His textbook ''Foundations of Analytic Number Theory'' went to two editions, 1975 and 1983. The Karatsuba algorithm is the earliest known divide and conquer algorithm for multiplication and lives on as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glyn Harman
Glyn Harman (born 2 November 1956) is a British mathematician working in analytic number theory. One of his major interests is prime number theory. He is best known for results on gaps between primes and the greatest prime factor of ''p'' + ''a'', as well as his lower bound for the number of Carmichael numbers up to X. His monograph ''Prime-detecting Sieves'' (2007) was published by Princeton University Press. He has also written a book ''Metric Number Theory'' (1998). As well, he has contributed to the field of Diophantine approximation. Harman also proved that there are infinitely many primes (additive primes) whose sum of digits is prime. (the sequencA046704in the OEIS).The OEIadditive primes/ref> Harman retired at the end of 2013 from being a professor at Royal Holloway, University of London. Previously he was a professor at Cardiff University. Harman is married, and has three sons, and used to live in Wokingham, Berkshire before moving to Harrow, Middlesex/Greater London ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedlander–Iwaniec Theorem
In analytic number theory the Friedlander–Iwaniec theorem states that there are infinitely many prime numbers of the form a^2 + b^4. The first few such primes are :2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, … . The difficulty in this statement lies in the very sparse nature of this sequence: the number of integers of the form a^2+b^4 less than X is roughly of the order X^. History The theorem was proved in 1997 by John Friedlander and Henryk Iwaniec. Iwaniec was awarded the 2001 Ostrowski Prize in part for his contributions to this work. Refinements The theorem was refined by D.R. Heath-Brown and Xiannan Li in 2017.. In particular, they proved that the polynomial a^2 + b^4 represents infinitely many primes when the variable b is also required to be prime. Namely, if f(n) is the prime numbers less then n in the form a^2 + b^4, then f(n) \sim v \frac where v=2 \sqrt \frac \prod_ \fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the 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 one in 2300 is prime ... [...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 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 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 method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brun Sieve
In the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Viggo Brun in 1915 and later generalized to the fundamental lemma of sieve theory by others. Description In terms of sieve theory the Brun sieve is of ''combinatorial type''; that is, it derives from a careful use of the inclusion–exclusion principle. Let A be a finite set of positive integers. Let P be some set of prime numbers. For each prime p in P, let A_p denote the set of elements of A that are divisible by p. This notation can be extended to other integers d that are products of distinct primes in P. In this case, define A_d to be the intersection of the sets A_p for the prime factors p of d. Finally, define A_1 to be A itself. Let z be an arbitrary positive real number. The object of the sieve is to estimate: S(A,P,z) = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sieve Theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit ''X''. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms. In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be. One successful approach is to approximate a specific sifted set of numbers (e.g. the set of prime numbers) by another, simpler set (e.g. the set of almost prime numbers), which is typically somewhat larger than the original set, and easier to analyze. More sophisticated sieves als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
