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The first deterministic primality test significantly faster than the naive methods was the cyclotomy test; its runtime can be proven to be O((log n) c log log log n), where n is the number to test for primality and c is a constant independent of n. Many further improvements were made, but none could be proven to have polynomial running time.
For this reason, strong primes are required by the ANSI X9.31 standard for use in generating RSA keys for digital signatures. However, strong primes do not protect against modulus factorisation using newer algorithms such as Lenstra elliptic curve factorization and Number Field Sieve algorithm.
The smallest such number is 25326001. This means that, if n is less than 25326001 and n is a strong probable prime to bases 2, 3, and 5, then n is prime. Carrying this further, 3825123056546413051 is the smallest number that is a strong pseudoprime to the 9 bases 2, 3, 5, 7, 11, 13, 17, 19, and 23.
Input #1: b, the number of bits of the result Input #2: k, the number of rounds of testing to perform Output: a strong probable prime n while True: pick a random odd integer n in the range [2 b −1 , 2 b −1] if the Miller–Rabin test with inputs n and k returns “ probably prime ” then return n
For example, \n does not represent a backslash followed by the letter n. The backslash escapes the compiler's normal, literal way of interpreting character data. After a backslash, the compiler expects subsequent characters to complete one of the defined escape sequences, and then translates the escape sequence into the characters it represents.
A Carmichael number will pass a Fermat primality test to every base relatively prime to the number, even though it is not actually prime. This makes tests based on Fermat's Little Theorem less effective than strong probable prime tests such as the Baillie–PSW primality test and the Miller–Rabin primality test.
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In 1930, Lev Schnirelmann proved that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant; see Schnirelmann density. [13] [14] Schnirelmann's constant is the lowest number C with this property. Schnirelmann himself obtained C < 800 000.