These are used in various applications, for example hashing, public-key cryptography, and search of prime factors in large numbers.
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.
A prime sieve works by creating a list of all integers up to a desired limit and progressively removing composite numbers (which it directly generates) until only primes are left.
This is the most efficient way to obtain a large range of primes; however, to find individual primes, direct primality tests are more efficient[citation needed].
Both the provable and probable primality tests rely on modular exponentiation.
To further reduce the computational cost, the integers are first checked for any small prime divisors using either sieves similar to the sieve of Eratosthenes or trial division.
In its usual standard implementation (which may include basic wheel factorization for small primes), it can find all the primes up to N in time
Note that just because an algorithm has decreased asymptotic time complexity does not mean that a practical implementation runs faster than an algorithm with a greater asymptotic time complexity: If in order to achieve that lesser asymptotic complexity the individual operations have a constant factor of increased time complexity that may be many times greater than for the simpler algorithm, it may never be possible within practical sieving ranges for the advantage of the reduced number of operations for reasonably large ranges to make up for this extra cost in time per operation.
Some sieving algorithms, such as the Sieve of Eratosthenes with large amounts of wheel factorization, take much less time for smaller ranges than their asymptotic time complexity would indicate because they have large negative constant offsets in their complexity and thus don't reach that asymptotic complexity until far beyond practical ranges.
For instance, the Sieve of Eratosthenes with a combination of wheel factorization and pre-culling using small primes up to 19 uses time of about a factor of two less than that predicted for the total range for a range of 1019, which total range takes hundreds of core-years to sieve for the best of sieve algorithms.
, which means that 1) they are very limited in the sieving ranges they can handle to the amount of RAM (memory) available and 2) that they are typically quite slow since memory access speed typically becomes the speed bottleneck more than computational speed once the array size grows beyond the size of the CPU caches.
The normally implemented page segmented sieves of both Eratosthenes and Atkin take space
plus small sieve segment buffers which are normally sized to fit within the CPU cache; page segmented wheel sieves including special variations of the Sieve of Eratosthenes typically take much more space than this by a significant factor in order to store the required wheel representations; Pritchard's variation of the linear time complexity sieve of Eratosthenes/wheel sieve takes
The better time complexity special version of the Sieve of Atkin takes space
Sorenson[4] shows an improvement to the wheel sieve that takes even less space at
However, the following is a general observation: the more the amount of memory is reduced, the greater the constant factor increase in the cost in time per operation even though the asymptotic time complexity may remain the same, meaning that the memory-reduced versions may run many times slower than the non-memory-reduced versions by quite a large factor.