Python library for primes algorithms

## Project description

## 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

## 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

### 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.

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