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The idea beneath this test is that when n is an odd prime, it passes the test because of two facts: by Fermat's little theorem, () (this property alone defines the weaker notion of probable prime to base a, on which the Fermat test is based); the only square roots of 1 modulo n are 1 and −1.
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.
Also, 2 is a prime dividing 100, which immediately proves that 100 is not prime. Every positive integer except 1 is divisible by at least one prime number by the Fundamental Theorem of Arithmetic . Therefore the algorithm need only search for prime divisors less than or equal to n {\displaystyle {\sqrt {n}}} .
Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 2 236 133 941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5 136 341 251. 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The numbers 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]
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