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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 tables contain the prime factorization of the natural numbers from 1 to 1000. When n is a prime number, the prime factorization is just n itself, written in bold below. The number 1 is called a unit. It has no prime factors and is neither prime nor composite.
As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that the probe sequence covers the whole table. [161] Some checksum methods are based on the mathematics of prime numbers.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [ 1 ] [ 2 ] The numbers p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89 . [ 3 ]
Another example is the distribution of the last digit of prime numbers. Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states that asymptotically, 25% of all primes end in each of these four digits.
The following table lists the progression of the largest known prime number in ascending order. [3] Here M p = 2 p − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was M 19 = 524,287, which was the largest known prime for 144 years. No records are known prior to 1456.
Consequently, a prime number divides at most one prime-exponent Mersenne number. [25] That is, the set of pernicious Mersenne numbers is pairwise coprime. If p and 2 p + 1 are both prime (meaning that p is a Sophie Germain prime ), and p is congruent to 3 (mod 4) , then 2 p + 1 divides 2 p − 1 .
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.