PrimeFinders

Installation

To install the package use pip:

pip install primefinders

Introduction

Some algorithm on prime numbers. You can find all the functions in the file primefinders/

In this release the Algorithm uses :

• Fermat's test (based on Fermat's theorem)

Future Algorithms will use :

• Eratosthenes sieve based
• Prime generating functions
• Miller-Rabin primality test
• Euler's phi(n) i.e., the number of integers less than n which have no common factor with n
• Lucas-Lehmer and LCG
• Atiyah's \Gamma

Specifications

• Language: Python 3.6 (also works on 3.5 and 3.7)
• Package:
  • Basic python packages were preferred
  • Matplotlib v2.0 - graph and math

Integration and pipeline

Code quality is monitored through codacity. For the tests coverage, there's codecov which is run during the Travis CI pipeline.

Math

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

Congruence

"≡" 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" 

See Modular_arithmetic

Strong Pseudoprime

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

Fermat's Theorem

How to use

A Probabilistic algorithm taking t randoms numbers a and testing the Fermat's theorem on number n > 1 Prime probability is right is 1 - 1/(2^t) Returns a boolean: True if n passes the tests.

With (n)

from primefinders import primefinders

primefinders.about()

primefinders.calculate(1697)

# With n the number you want to test
fermat(n)
primefinders.fermat(1697)

Theory

If n is prime then ∀ a ∈[1, ..., n-1]

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

Lucas-Lehmer

How to use

Implementation of the sieve of erathostenes that discover the primes and their composite up to a limit. It returns a dictionary:

• the key are the primes up to n
• the value is the list of composites of these primes up to n

from primefinders import toolkit
>>>

from primefinders import lucas_lehmer

# With as a parameter the upper limit
lucas_lehmer(10)
>>> 

Sieve of Eratosthenes

How to use

Implementation of the sieve of erathostenes that discover the primes and their composite up to a limit. It returns a dictionary:

• the key are the primes up to n
• the value is the list of composites of these primes up to n

from primefinders import sieve_eratosthenes

# With as a parameter the upper limit
sieve_eratosthenes(10)
>> {2: [4, 6, 8], 3: [6, 9], 5: [], 7: []}

Theory

This sieve mark as composite the multiple of each primes. It is an efficient way to find primes. For n ∈ N with n > 2 and for ∀ a ∈[2, ..., √n] then n/a ∉ N is true.

Atiyah's

How to use