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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). Dickson's conjecture is further extended by Schinzel's hypothesis H. 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 infi ... [...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] |
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] |
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] |
Dirichlet's Theorem On Arithmetic Progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is also a positive integer. In other words, there are infinitely many primes that are congruent to ''a'' modulo ''d''. The numbers of the form ''a'' + ''nd'' form an arithmetic progression :a,\ a+d,\ a+2d,\ a+3d,\ \dots,\ and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem, named after Peter Gustav Lejeune Dirichlet, extends Euclid's theorem that there are infinitely many prime numbers. Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the ... [...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 (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] |
Sophie Germain Prime
In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. One attempt by Germain to prove Fermat’s Last Theorem was to let ''p'' be a prime number of the form 8''k'' + 7 and to let ''n'' = ''p'' – 1. In this case, x^n + y^n = z^n is unsolvable. Germain’s proof, however, remained unfinished. Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theorem which states that if ''p'' is an odd prime and 2''p'' + 1 is also prime, then ''p'' must divide ''x'', ''y'', or ''z.'' Otherwise, x^n + y^n \neq z^n. This case where ''p'' does not divide ''x'', ''y'', or ''z'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schinzel's Hypothesis H
In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalization of widely open conjectures such as the twin prime conjecture. The hypothesis is named after Andrzej Schinzel. Statement The hypothesis claims that for every finite collection \ of nonconstant irreducible polynomials over the integers with positive leading coefficients, ''one of the following conditions'' holds: # There are infinitely many positive integers n such that all of f_1(n),f_2(n),\ldots,f_k(n) are simultaneously prime numbers, or # There is an integer m>1 (called a ''fixed divisor'') which always divides the product f_1(n)f_2(n)\cdots f_k(n). (Or, equivalently: There exists a prime p such that for every n there is an i such that p divides f_i(n).) The second condition is satisfied by sets such as f_1(x)=x+4, f_2(x)=x+7, since (x+4)(x+7) is always divisible by 2. It is easy to see that this condition prevents the ''first'' condit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bunyakovsky Conjecture
The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial f(x) in one variable with integer coefficients to give infinitely many prime values in the sequencef(1), f(2), f(3),\ldots. It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky. The following three conditions are necessary for f(x) to have the desired prime-producing property: # The leading coefficient is positive, # The polynomial is irreducible over the rationals (and integers). # The values f(1), f(2), f(3),\ldots have no common factor. (In particular, the coefficients of f(x) should be relatively prime.) Bunyakovsky's conjecture is that these conditions are sufficient: if f(x) satisfies (1)–(3), then f(n) is prime for infinitely many positive integers n. A seemingly weaker yet equivalent statement to Bunyakovsky's conjecture is that for every integer polynomial f(x) that satisfies (1)–(3), f(n) is prime for ''at least one'' positive integer n: but then, since th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Triplet
In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself). Examples The first prime triplets are (5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887) Subpairs of primes A prime triplet contains a single pair of: *Twin primes: or ; *Cousin p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green–Tao Theorem
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number ''k'', there exist arithmetic progressions of primes with ''k'' terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.. Statement Let \pi(N) denote the number of primes less than or equal to N. If A is a subset of the prime numbers such that : \limsup_ \frac>0, then for all positive integers k, the set A contains infinitely many arithmetic progressions of length k. In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions. In their later work on the generalized Hardy–Littlewood conjecture, Green and Tao stated and conditionally proved the asymptotic formula : (\mathfrak_k + o(1))\frac for the number of ''k'' tuples of primes ... [...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] |
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] |
