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In number theory, Dirichlet's theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n > 0, or in other words: there are infinitely many primes which are congruent to a modulo d. Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ...
In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself. ...
In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...
Modular arithmetic is a modified system of arithmetic for integers, sometimes referred to as clock arithmetic, where numbers wrap around after they reach a certain value (the modulus). ...
This theorem extends Euclid's theorem that there are infinitely many primes (in this case of the form 3 + 4n, which are also the Gaussian primes, or of the form 1 + 2n, for every odd number, excluding 1). Note that the theorem does not say that there are infinitely many consecutive terms in the arithmetic progression Euclid of Alexandria (Greek: ) (circa 365–275 BC) was a Greek mathematician, now known as the father of geometry. His most famous work is Elements, widely considered to be historys most successful textbook. ...
A Gaussian integer is a complex number whose real and imaginary part are both integers. ...
In mathematics, any integer (whole number) is either even or odd. ...
One redirects here. ...
In mathematics, an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ...
- a, a+d, a+2d, a+3d, ...,
which are prime. For example in we get primes of the type 4n + 3 'only' for n with the values - 1, 2, 4, 5, 7, 10, 11, 14, 16, 17, 19, 20, 25, 26, 31, 32, 34, 37, 40, 41, 44, 47, 49, 52, 55, 56, 59, 62, 65, 67, 70, 76, 77, 82, 86, 89, 91, 94, 95, ... .
In fact since the primes thin out, on average, the same must be true for the primes in arithmetic progressions. One naturally then asks about the way the primes are shared between the various arithmetic progressions for a given value of d (there are d of those, essentially, if we don't distinguish two progressions sharing almost all their terms). The answer is given in this form: the number of feasible progressions modulo d — those that are not ruled out on the grounds that a and d have a common factor > 1 — is given by Euler's totient function In mathematics, the phrase almost all has a number of specialised uses. ...
In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...
- φ(d).
Further, the natural density of primes in each of those is - 1/φ(d).
For example if d is a prime number q, each of the q − 1 progressions, other than - q, 2q, 3q, ...
contains a proportion 1/(q − 1) of the primes.
History
Euler stated that every arithmetic progression beginning with 1 contains an infinite number of primes. The theorem in the above form was first conjectured by Gauss and proved by Dirichlet in 1835 with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the distribution of primes. The theorem represents the beginning of rigorous analytic number theory. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...
Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) ( April 30, 1777 - February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. ...
Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ...
1835 was a common year starting on Thursday (see link for calendar). ...
In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ...
In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ...
Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ...
In algebraic number theory Dirichlet's theorem generalizes to Chebotarev's density theorem. In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...
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