Which is the fastest algorithm to find prime numbers?

Question

A very fast implementation of the Sieve of Atkin is Dan Bernstein's primegen. This sieve is more efficient than the Sieve of Eratosthenes. His page has some benchmark information.

If it has to be really fast you can include a list of primes:
http://www.bigprimes.net/archive/prime/

If you just have to know if a certain number is a prime number, there are various prime tests listed on wikipedia. They are probably the fastest method to determine if large numbers are primes, especially because they can tell you if a number is not a prime.

He, he I know I'm a question necromancer replying to old questions, but I've just found this question searching the net for ways to implement efficient prime numbers tests.

Until now, I believe that the fastest prime number testing algorithm is Strong Probable Prime (SPRP). I am quoting from Nvidia CUDA forums:

One of the more practical niche problems in number theory has to do with identification of prime numbers. Given N, how can you efficiently determine if it is prime or not? This is not just a thoeretical problem, it may be a real one needed in code, perhaps when you need to dynamically find a prime hash table size within certain ranges. If N is something on the order of 2^30, do you really want to do 30000 division tests to search for any factors? Obviously not.

The common practical solution to this problem is a simple test called an Euler probable prime test, and a more powerful generalization called a Strong Probable Prime (SPRP). This is a test that for an integer N can probabilistically classify it as prime or not, and repeated tests can increase the correctness probability. The slow part of the test itself mostly involves computing a value similar to A^(N-1) modulo N. Anyone implementing RSA public-key encryption variants has used this algorithm. It's useful both for huge integers (like 512 bits) as well as normal 32 or 64 bit ints.

The test can be changed from a probabilistic rejection into a definitive proof of primality by precomputing certain test input parameters which are known to always succeed for ranges of N. Unfortunately the discovery of these "best known tests" is effectively a search of a huge (in fact infinite) domain. In 1980, a first list of useful tests was created by Carl Pomerance (famous for being the one to factor RSA-129 with his Quadratic Seive algorithm.) Later Jaeschke improved the results significantly in 1993. In 2004, Zhang and Tang improved the theory and limits of the search domain. Greathouse and Livingstone have released the most modern results until now on the web, at http://math.crg4.com/primes.html, the best results of a huge search domain.

See here for more info: http://primes.utm.edu/prove/prove2_3.html and http://forums.nvidia.com/index.php?showtopic=70483

If you just need a way to generate very big prime numbers and don't care to generate all prime numbers < an integer n, you can use Lucas-Lehmer test to verify Mersenne prime numbers. A Mersenne prime number is in the form of 2^p -1. I think that Lucas-Lehmer test is the fastest algorithm discovered for Mersenne prime numbers.

And if you not only want to use the fastest algorithm but also the fastest hardware, try to implement it using Nvidia CUDA, write a kernel for CUDA and run it on GPU.

You can even earn some money if you discover large enough prime numbers, EFF is giving prizes from $50K to $250K: https://www.eff.org/awards/coop

There is a 100% mathematical test that will check if a number P is prime or composite, called AKS Primality Test.

The concept is simple: given a number P, if all the coefficients of (x-1)^P - (x^P-1) are divisible by P, then P is a prime number, otherwise it is a composite number.

For instance, given P = 3, would give the polynomial:

   (x-1)^3 - (x^3 - 1)
 = x^3 + 3x^2 - 3x - 1 - (x^3 - 1)
 = 3x^2 - 3x

And the coefficients are both divisible by 3, therefore the number is prime.

And example where P = 4, which is NOT a prime would yield:

   (x-1)^4 - (x^4-1)
 = x^4 - 4x^3 + 6x^2 - 4x + 1 - (x^4 - 1)
 = -4x^3 + 6x^2 - 4x

And here we can see that the coefficients 6 is not divisible by 4, therefore it is NOT prime.

The polynomial (x-1)^P will P+1 terms and can be found using combination. So, this test will run in O(n) runtime, so I don't know how useful this would be since you can simply iterate over i from 0 to p and test for the remainder.

Is your problem to decide whether a particular number is prime? Then you need a primality test (easy). Or do you need all primes up to a given number? In that case prime sieves are good (easy, but require memory). Or do you need the prime factors of a number? This would require factorization (difficult for large numbers if you really want the most efficient methods). How large are the numbers you are looking at? 16 bits? 32 bits? bigger?

One clever and efficient way is to pre-compute tables of primes and keep them in a file using a bit-level encoding. The file is considered one long bit vector whereas bit n represents integer n. If n is prime, its bit is set to one and to zero otherwise. Lookup is very fast (you compute the byte offset and a bit mask) and does not require loading the file in memory.