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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.
Also, 2 is a prime dividing 100, which immediately proves that 100 is not prime. Every positive integer except 1 is divisible by at least one prime number by the Fundamental Theorem of Arithmetic . Therefore the algorithm need only search for prime divisors less than or equal to n {\displaystyle {\sqrt {n}}} .
The PrimePages is a website about prime numbers originally created by Chris Caldwell at the University of Tennessee at Martin [2] who maintained it from 1994 to 2023.. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms.
A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially, let p equal 2, the smallest prime number.
The sieve starts with a list of the integers from 1 to n. From this list, all numbers of the form i + j + 2ij are removed, where i and j are positive integers such that 1 ≤ i ≤ j and i + j + 2ij ≤ n. The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (that is, all primes except 2) below 2n + 2.
The numbers which remain prime under cyclic shifts of digits. A016114: Home prime: 1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ... For n ≥ 2, a(n) is the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached; a(n) = −1 if no prime is ever reached. A037274
Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers. A composite number has Ω(n) > 1.
For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. The next larger unique prime is 9091 with period 10, though the next larger period is 9 (its prime being 333667).