README.md

nprime

Installation

To install the package use pip:

pip install nprime

Introduction

Some algorithm on prime numbers. You can find all the functions in the file nprime/pryprime.py

Algorithm developed :

• Eratosthenes sieve based
• Fermat's test (based on Fermat's theorem)
• Prime generating functions
• Miller Rabin predictive algorithm

Specifications

• Language: Python 3.5.2
• Package:
  • Basic python packages were preferred
  • Matplotlib v2.1 - 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" 

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

Erathostene's Sieve

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 nprime import sieve_eratosthenes

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

The previous behaviour can be called using the trial_division which uses the Trial Division algorithm

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.

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.

from nprime import fermat

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

Theory

If n is prime then ∀ a ∈[1, ..., n-1]

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

Miller rabin

How to use

A probabilistic algorithm which determines whether a given number (n > 1) is prime or not. The miller_rabin tests is repeated t times to get more accurate results. Returns a boolean: True if n passes the tests.

from nprime import miller_rabin

# With n the number you want to test
miller_rabin(n)

Theory

For n ∈ N and n > 2,
Take a random a ∈ {1,...,n−1}
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.