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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.
In the Parsons problem exam question, the first example shown, there are 6 pairs of statements, one mark is deducted for each incorrectly chosen line from the pairs, two marks are deducted if more than 2 lines are out of place, and one mark is deducted if up to two things are out of place.
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.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89. [3]
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
All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.
For example, take n = 71. Then n − 1 = 70 and the prime factors of 70 are 2, 5 and 7.We randomly select an a=17 < n.Now we compute: (). For all integers a it is known that
Using repeated squaring, the running time of this algorithm is O(k n 3), for an n-digit number, and k is the number of rounds performed; thus this is an efficient, polynomial-time algorithm. FFT-based multiplication, for example the Schönhage–Strassen algorithm, can decrease the running time to O(k n 2 log n log log n) = Õ(k n 2).