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A primality test is an algorithm for determining whether an input number is prime.Among other fields of mathematics, it is used for cryptography.Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not.
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
This must always hold if n is prime; if not, we have found more than two square roots of −1 and proved that n is composite. This is only possible if n ≡ 1 (mod 4), and we pass probable prime tests with two or more bases a such that a d ≢ ±1 (mod n), but it is an inexpensive addition to the basic Miller-Rabin test.
Check if n is a perfect power: if n = a b for integers a > 1 and b > 1, then output composite. Find the smallest r such that ord r (n) > (log 2 n) 2. If r and n are not coprime, then output composite. For all 2 ≤ a ≤ min (r, n−1), check that a does not divide n: If a|n for some 2 ≤ a ≤ min (r, n−1), then output composite.
So we still don't know if 71 is prime or not. We try another random a, this time choosing a = 11. Now we compute: (). Again, this does not show that the multiplicative order of 11 (mod 71) is 70 because some factor of 70 may also work. So check 70 divided by its prime factors:
Otherwise, p is a strong probable prime to base a; that is, it may be prime or not. If p is composite, the probability that the test declares it a strong probable prime anyway is at most 1 ⁄ 4, in which case p is a strong pseudoprime, and a is a strong liar.
Mark as non-prime the positions in the array corresponding to the multiples of each prime p ≤ √ m found so far, by enumerating its multiples in steps of p starting from the lowest multiple of p between m - Δ and m. The remaining non-marked positions in the array correspond to the primes in the segment.
The Mersenne number M 3 = 2 3 −1 = 7 is prime. The Lucas–Lehmer test verifies this as follows. Initially s is set to 4 and then is updated 3−2 = 1 time: s ← ((4 × 4) − 2) mod 7 = 0. Since the final value of s is 0, the conclusion is that M 3 is prime. On the other hand, M 11 = 2047 = 23 × 89 is not prime