Generation of primes

In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently. These are used in various applications, for example hashing, public-key cryptography, and search of prime factors in large numbers.

For relatively small numbers, it is possible to just apply trial division to each successive odd number. Prime sieves are almost always faster. Prime sieving is the fastest known way to deterministically enumerate the primes. There are some known formulas that can calculate the next prime but there is no known way to express the next prime in terms of the previous primes. Also, there is no effective known general manipulation and/or extension of some mathematical expression (even such including later primes) that deterministically calculates the next prime.