Dickson's conjecture
Encyclopedia
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 special cases are well known conjectures: there are infinitely many twin prime
s (n and n + 2 are primes), and there are infinitely many Sophie Germain prime
s (n and 2n + 1 are primes).
Dickson's conjecture is further extended by Schinzel's hypothesis H
.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, 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
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, unless there is a congruence condition preventing this . The case k = 1 is Dirichlet's theorem
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 ≥ 0. In other words, there are infinitely many primes which are...
.
Two special cases are well known conjectures: there are infinitely many twin prime
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...
s (n and n + 2 are primes), and there are infinitely many Sophie Germain prime
Sophie Germain prime
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number...
s (n and 2n + 1 are primes).
Dickson's conjecture is further extended by Schinzel's hypothesis H
Schinzel's hypothesis H
In mathematics, Schinzel's hypothesis H is a very broad generalisation of conjectures such as the twin prime conjecture. It aims to define the possible scope of a conjecture of the nature that several sequences of the type...
.