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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 extends Euclid's theorem that there are infinitely many prime numbers (of the form 1 + 2n). 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 primes are evenly dis ... [...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] |
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. He is also known for his contributions to the Least squares, method of least squares, and was the first to officially publish on it, though Carl Friedrich Gauss had discovered it before him. Life Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale Supérieure, École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Prussian Academy of Sciences, Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant m ... [...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] |
Open Problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is known). In the history of science, some of these supposed open problems were "solved" by means of showing that they were not well-defined. In mathematics, many open problems are concerned with the question of whether a certain definition is or is not consistent. Two notable examples in mathematics that have been solved and ''closed'' by researchers in the late twentieth century are Fermat's Last Theorem and the four-color theorem.K. Appel and W. Haken (1977), "Every planar map is four colorable. Part I. Discharging", ''Illinois J. Math'' 21: 429–490. K. Appel, W. Haken, and J. Koch (1977), "Every planar map is four colorable. Part II. Reducibility", ''Illinois J. Math'' 21: 491–567. An important open mathematics problem solved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landau's Problems
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows: # Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes? # Twin prime conjecture: Are there infinitely many primes ''p'' such that ''p'' + 2 is prime? # Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares? # Are there infinitely many primes ''p'' such that ''p'' − 1 is a perfect square? In other words: Are there infinitely many primes of the form ''n''2 + 1? , all four problems are unresolved. Progress toward solutions Goldbach's conjecture Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Gold ... [...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), and # There is no common factor for all the infinitely many values f(1), f(2), f(3),\ldots. (In particular, the coefficients of f(x) should be relatively prime. It is not necessary for the values f(n) to be pairwise 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 polyno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi: \mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: # \chi(ab) = \chi(a)\chi(b); that is, \chi is completely multiplicative. # \chi(a) \begin =0 &\text \gcd(a,m)>1\\ \ne 0&\text\gcd(a,m)=1. \end (gcd is the greatest common divisor) # \chi(a + m) = \chi(a); that is, \chi is periodic with period m. The simplest possible character, called the principal character, usually denoted \chi_0, (see Notation below) exists for all moduli: : \chi_0(a)= \begin 0 &\text \gcd(a,m)>1\\ 1 &\text \gcd(a,m)=1. \end The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions. Notation \phi(n) is Euler's totient function. \zeta_n is a complex primitive n-th root of unity: : \zeta_n^n=1, but \zeta_n\ne 1, \ze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet L-function
In mathematics, a Dirichlet L-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and s a complex variable with real part greater than 1 . It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L -function and also denoted L ( s , \chi) . These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the Dirichlet's theorem on arithmetic progressions, theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L ( s , \chi) is non-zero at s = 1 . Moreover, if \chi is principal, then the corresponding Dirichlet L -function has a simple pole at s = 1 . Otherwise, the L -function is entire function, entire. Euler product Since a Dirichlet character \chi is completely multiplicative, its L -function can also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Proof
In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain theorems, like the prime number theorem, could only be proved by invoking "higher" mathematical theorems or techniques. However, as time progresses, many of these results have also been subsequently reproven using only elementary techniques. While there is generally no consensus as to what counts as elementary, the term is nevertheless a common part of the mathematical jargon. An elementary proof is not necessarily simple, in the sense of being easy to understand or trivial. In fact, some elementary proofs can be quite complicated — and this is especially true when a statement of notable importance is involved.. Prime number theorem The distinction between elementary and non-elementary proofs has been considered especially important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. * Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet L-series
In mathematics, a Dirichlet L-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and s a complex variable with real part greater than 1 . It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L -function and also denoted L ( s , \chi) . These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L ( s , \chi) is non-zero at s = 1 . Moreover, if \chi is principal, then the corresponding Dirichlet L -function has a simple pole at s = 1 . Otherwise, the L -function is entire. Euler product Since a Dirichlet character \chi is completely multiplicative, its L -function can also be written as an Euler product in the half-plane of absolute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
