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In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise.
Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
The prime number theorem is obtained there in an equivalent form that the Cesàro sum of the values of the Liouville function is zero. The Liouville function is ( − 1 ) ω ( n ) {\displaystyle (-1)^{\omega (n)}} where ω ( n ) {\displaystyle \omega (n)} is the number of prime factors, with multiplicity, of the integer n {\displaystyle n} .
A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.
Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers.
A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and ... Partition function values that are prime. 2, 3, 5, 7, ...
In 1737, Euler related the study of prime numbers to what is known now as the Riemann zeta function: he showed that the value () reduces to a ratio of two infinite products, Π p / Π (p–1), for all primes p, and that the ratio is infinite. [1] [2] In 1775, Euler stated the theorem for the cases of a + nd, where a = 1. [3]
Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). [4] [5] This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring /). [6]