Some algorythm on prime numbers.

Language: Python 3.5.2

Algorithm developped :

• Native one (prime through divisions)
• Eratosthenes sieve based
• Fermat's test (based on Fermat's theorem)
• Prime generating functions

Here are a bit of information to help understand some of the algorithms

"≡" means congruent, a ≡ b (mod m) implies that m / (a-b), ∃ k ∈ Z that verifies a = kn + b

which implies:

a ≡ 0 (mod n) <-> a = kn <-> "a" is divisible by "n" 

if n is prime then ∀ a ∈[1, ..., n-1]

a^(n-1) ≡ 1 (mod n) ⇔ a^(n-1) = kn + 1

Take a random a ∈ {1,...,n−1} and n > 2,
Find d and s such as with n - 1 = 2^s * d (with d odd)
if (a^d)^2^r ≡ 1 mod n for all r in 0 to s-1
Then n is prime.

The test output is false of 1/4 of the "a values" possible in n, so the test is repeated t times.

A strong pseudoprime to a base a is an odd composite number n with n-1 = d·2^s (for d odd) for which either a^d = 1(mod n) or a^(d·2^r) = -1(mod n) for some r = 0, 1, ..., s-1