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This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.
However, it does not contain all the prime numbers, since the terms gcd(n + 1, a n) are always odd and so never equal to 2. 587 is the smallest prime (other than 2) not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper it was conjectured to contain all odd primes, even though it is rather inefficient.
The prime number race generalizes to other moduli and is the subject of much research; Pál Turán asked whether it is always the case that π c,a (x) and π c,b (x) change places when a and b are coprime to c. [34]
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2 p−1 × (2 p − 1), where 2 p − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form.
Sequences dn + a with odd d are often ignored because half the numbers are even and the other half is the same numbers as a sequence with 2d, if we start with n = 0. For example, 6n + 1 produces the same primes as 3n + 1, while 6n + 5 produces the same as 3n + 2 except for the only even prime 2. The following table lists several arithmetic ...
See List of prime numbers for definitions and examples of many classes of primes. Pages in category "Classes of prime numbers" The following 76 pages are in this category, out of 76 total.