Search results
Results From The WOW.Com Content Network
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.
This theorem is a consequence of the pigeonhole principle. Peter Gustav Lejeune Dirichlet who proved the result used the same principle in other contexts (for example, the Pell equation) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later. [2]
Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory. [15] He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem.
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. [1] 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.
Dirichlet's theorem may refer to any of several mathematical theorems due to Peter Gustav Lejeune Dirichlet. Dirichlet's theorem on arithmetic progressions;
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. [1] It determines the rank of the group of units in the ring O K of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.
Dirichlet's unit theorem holds for S-units: the group of S-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to r + s, where r is the rank of the unit group and s = |S|.
Dirichlet 1. Dirichlet's theorem on arithmetic progressions 2. Dirichlet character 3. Dirichlet's unit theorem. distribution A distribution in number theory is a generalization/variant of a distribution in analysis. divisor A divisor or factor of an integer n is an integer m such that there exists an integer k satisfying n = mk. Divisors can be ...