#### Primelib

p
```"""
Created on Thu Oct  5 16:44:23 2017

@author: Christian Bender

This Python library contains some useful functions to deal with
prime numbers and whole numbers.

Overview:

is_prime(number)
sieve_er(N)
get_prime_numbers(N)
prime_factorization(number)
greatest_prime_factor(number)
smallest_prime_factor(number)
get_prime(n)
get_primes_between(pNumber1, pNumber2)

----

is_even(number)
is_odd(number)
kg_v(number1, number2)  // least common multiple
get_divisors(number)    // all divisors of 'number' inclusive 1, number
is_perfect_number(number)

NEW-FUNCTIONS

simplify_fraction(numerator, denominator)
factorial (n) // n!
fib (n) // calculate the n-th fibonacci term.

-----

goldbach(number)  // Goldbach's assumption

"""

from math import sqrt

from maths.greatest_common_divisor import gcd_by_iterative

def is_prime(number: int) -> bool:
"""
input: positive integer 'number'
returns true if 'number' is prime otherwise false.

>>> is_prime(3)
True
>>> is_prime(10)
False
>>> is_prime(97)
True
>>> is_prime(9991)
False
>>> is_prime(-1)
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and positive
>>> is_prime("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and positive
"""

# precondition
assert isinstance(number, int) and (
number >= 0
), "'number' must been an int and positive"

status = True

# 0 and 1 are none primes.
if number <= 1:
status = False

for divisor in range(2, int(round(sqrt(number))) + 1):
# if 'number' divisible by 'divisor' then sets 'status'
# of false and break up the loop.
if number % divisor == 0:
status = False
break

# precondition
assert isinstance(status, bool), "'status' must been from type bool"

return status

# ------------------------------------------

def sieve_er(n):
"""
input: positive integer 'N' > 2
returns a list of prime numbers from 2 up to N.

This function implements the algorithm called
sieve of erathostenes.

>>> sieve_er(8)
[2, 3, 5, 7]
>>> sieve_er(-1)
Traceback (most recent call last):
...
AssertionError: 'N' must been an int and > 2
>>> sieve_er("test")
Traceback (most recent call last):
...
AssertionError: 'N' must been an int and > 2
"""

# precondition
assert isinstance(n, int) and (n > 2), "'N' must been an int and > 2"

# beginList: contains all natural numbers from 2 up to N
begin_list = list(range(2, n + 1))

ans = []  # this list will be returns.

# actual sieve of erathostenes
for i in range(len(begin_list)):
for j in range(i + 1, len(begin_list)):
if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
begin_list[j] = 0

# filters actual prime numbers.
ans = [x for x in begin_list if x != 0]

# precondition
assert isinstance(ans, list), "'ans' must been from type list"

return ans

# --------------------------------

def get_prime_numbers(n):
"""
input: positive integer 'N' > 2
returns a list of prime numbers from 2 up to N (inclusive)
This function is more efficient as function 'sieveEr(...)'

>>> get_prime_numbers(8)
[2, 3, 5, 7]
>>> get_prime_numbers(-1)
Traceback (most recent call last):
...
AssertionError: 'N' must been an int and > 2
>>> get_prime_numbers("test")
Traceback (most recent call last):
...
AssertionError: 'N' must been an int and > 2
"""

# precondition
assert isinstance(n, int) and (n > 2), "'N' must been an int and > 2"

ans = []

# iterates over all numbers between 2 up to N+1
# if a number is prime then appends to list 'ans'
for number in range(2, n + 1):
if is_prime(number):
ans.append(number)

# precondition
assert isinstance(ans, list), "'ans' must been from type list"

return ans

# -----------------------------------------

def prime_factorization(number):
"""
input: positive integer 'number'
returns a list of the prime number factors of 'number'

>>> prime_factorization(0)
[0]
>>> prime_factorization(8)
[2, 2, 2]
>>> prime_factorization(287)
[7, 41]
>>> prime_factorization(-1)
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and >= 0
>>> prime_factorization("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and >= 0
"""

# precondition
assert isinstance(number, int) and number >= 0, "'number' must been an int and >= 0"

ans = []  # this list will be returns of the function.

# potential prime number factors.

factor = 2

quotient = number

if number in {0, 1}:
ans.append(number)

# if 'number' not prime then builds the prime factorization of 'number'
elif not is_prime(number):
while quotient != 1:
if is_prime(factor) and (quotient % factor == 0):
ans.append(factor)
quotient /= factor
else:
factor += 1

else:
ans.append(number)

# precondition
assert isinstance(ans, list), "'ans' must been from type list"

return ans

# -----------------------------------------

def greatest_prime_factor(number):
"""
input: positive integer 'number' >= 0
returns the greatest prime number factor of 'number'

>>> greatest_prime_factor(0)
0
>>> greatest_prime_factor(8)
2
>>> greatest_prime_factor(287)
41
>>> greatest_prime_factor(-1)
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and >= 0
>>> greatest_prime_factor("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and >= 0
"""

# precondition
assert isinstance(number, int) and (
number >= 0
), "'number' must been an int and >= 0"

ans = 0

# prime factorization of 'number'
prime_factors = prime_factorization(number)

ans = max(prime_factors)

# precondition
assert isinstance(ans, int), "'ans' must been from type int"

return ans

# ----------------------------------------------

def smallest_prime_factor(number):
"""
input: integer 'number' >= 0
returns the smallest prime number factor of 'number'

>>> smallest_prime_factor(0)
0
>>> smallest_prime_factor(8)
2
>>> smallest_prime_factor(287)
7
>>> smallest_prime_factor(-1)
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and >= 0
>>> smallest_prime_factor("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and >= 0
"""

# precondition
assert isinstance(number, int) and (
number >= 0
), "'number' must been an int and >= 0"

ans = 0

# prime factorization of 'number'
prime_factors = prime_factorization(number)

ans = min(prime_factors)

# precondition
assert isinstance(ans, int), "'ans' must been from type int"

return ans

# ----------------------

def is_even(number):
"""
input: integer 'number'
returns true if 'number' is even, otherwise false.

>>> is_even(0)
True
>>> is_even(8)
True
>>> is_even(287)
False
>>> is_even(-1)
False
>>> is_even("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int
"""

# precondition
assert isinstance(number, int), "'number' must been an int"
assert isinstance(number % 2 == 0, bool), "compare must been from type bool"

return number % 2 == 0

# ------------------------

def is_odd(number):
"""
input: integer 'number'
returns true if 'number' is odd, otherwise false.

>>> is_odd(0)
False
>>> is_odd(8)
False
>>> is_odd(287)
True
>>> is_odd(-1)
True
>>> is_odd("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int
"""

# precondition
assert isinstance(number, int), "'number' must been an int"
assert isinstance(number % 2 != 0, bool), "compare must been from type bool"

return number % 2 != 0

# ------------------------

def goldbach(number):
"""
Goldbach's assumption
input: a even positive integer 'number' > 2
returns a list of two prime numbers whose sum is equal to 'number'

>>> goldbach(8)
[3, 5]
>>> goldbach(824)
[3, 821]
>>> goldbach(0)
Traceback (most recent call last):
...
AssertionError: 'number' must been an int, even and > 2
>>> goldbach(-1)
Traceback (most recent call last):
...
AssertionError: 'number' must been an int, even and > 2
>>> goldbach("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int, even and > 2
"""

# precondition
assert (
isinstance(number, int) and (number > 2) and is_even(number)
), "'number' must been an int, even and > 2"

ans = []  # this list will returned

# creates a list of prime numbers between 2 up to 'number'
prime_numbers = get_prime_numbers(number)
len_pn = len(prime_numbers)

# run variable for while-loops.
i = 0
j = None

# exit variable. for break up the loops
loop = True

while i < len_pn and loop:
j = i + 1

while j < len_pn and loop:
if prime_numbers[i] + prime_numbers[j] == number:
loop = False
ans.append(prime_numbers[i])
ans.append(prime_numbers[j])

j += 1

i += 1

# precondition
assert (
isinstance(ans, list)
and (len(ans) == 2)
and (ans[0] + ans[1] == number)
and is_prime(ans[0])
and is_prime(ans[1])
), "'ans' must contains two primes. And sum of elements must been eq 'number'"

return ans

# ----------------------------------------------

def kg_v(number1, number2):
"""
Least common multiple
input: two positive integer 'number1' and 'number2'
returns the least common multiple of 'number1' and 'number2'

>>> kg_v(8,10)
40
>>> kg_v(824,67)
55208
>>> kg_v(1, 10)
10
>>> kg_v(0)
Traceback (most recent call last):
...
TypeError: kg_v() missing 1 required positional argument: 'number2'
>>> kg_v(10,-1)
Traceback (most recent call last):
...
AssertionError: 'number1' and 'number2' must been positive integer.
>>> kg_v("test","test2")
Traceback (most recent call last):
...
AssertionError: 'number1' and 'number2' must been positive integer.
"""

# precondition
assert (
isinstance(number1, int)
and isinstance(number2, int)
and (number1 >= 1)
and (number2 >= 1)
), "'number1' and 'number2' must been positive integer."

ans = 1  # actual answer that will be return.

# for kgV (x,1)
if number1 > 1 and number2 > 1:
# builds the prime factorization of 'number1' and 'number2'
prime_fac_1 = prime_factorization(number1)
prime_fac_2 = prime_factorization(number2)

elif number1 == 1 or number2 == 1:
prime_fac_1 = []
prime_fac_2 = []
ans = max(number1, number2)

count1 = 0
count2 = 0

done = []  # captured numbers int both 'primeFac1' and 'primeFac2'

# iterates through primeFac1
for n in prime_fac_1:
if n not in done:
if n in prime_fac_2:
count1 = prime_fac_1.count(n)
count2 = prime_fac_2.count(n)

for _ in range(max(count1, count2)):
ans *= n

else:
count1 = prime_fac_1.count(n)

for _ in range(count1):
ans *= n

done.append(n)

# iterates through primeFac2
for n in prime_fac_2:
if n not in done:
count2 = prime_fac_2.count(n)

for _ in range(count2):
ans *= n

done.append(n)

# precondition
assert isinstance(ans, int) and (
ans >= 0
), "'ans' must been from type int and positive"

return ans

# ----------------------------------

def get_prime(n):
"""
Gets the n-th prime number.
input: positive integer 'n' >= 0
returns the n-th prime number, beginning at index 0

>>> get_prime(0)
2
>>> get_prime(8)
23
>>> get_prime(824)
6337
>>> get_prime(-1)
Traceback (most recent call last):
...
AssertionError: 'number' must been a positive int
>>> get_prime("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been a positive int
"""

# precondition
assert isinstance(n, int) and (n >= 0), "'number' must been a positive int"

index = 0
ans = 2  # this variable holds the answer

while index < n:
index += 1

ans += 1  # counts to the next number

# if ans not prime then
# runs to the next prime number.
while not is_prime(ans):
ans += 1

# precondition
assert isinstance(ans, int) and is_prime(
ans
), "'ans' must been a prime number and from type int"

return ans

# ---------------------------------------------------

def get_primes_between(p_number_1, p_number_2):
"""
input: prime numbers 'pNumber1' and 'pNumber2'
pNumber1 < pNumber2
returns a list of all prime numbers between 'pNumber1' (exclusive)
and 'pNumber2' (exclusive)

>>> get_primes_between(3, 67)
[5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61]
>>> get_primes_between(0)
Traceback (most recent call last):
...
TypeError: get_primes_between() missing 1 required positional argument: 'p_number_2'
>>> get_primes_between(0, 1)
Traceback (most recent call last):
...
AssertionError: The arguments must been prime numbers and 'pNumber1' < 'pNumber2'
>>> get_primes_between(-1, 3)
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and positive
>>> get_primes_between("test","test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and positive
"""

# precondition
assert (
is_prime(p_number_1) and is_prime(p_number_2) and (p_number_1 < p_number_2)
), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

ans = []  # this list will be returns.

# if number is not prime then
# fetch the next prime number.
while not is_prime(number):
number += 1

while number < p_number_2:
ans.append(number)

number += 1

# fetch the next prime number.
while not is_prime(number):
number += 1

# precondition
assert (
isinstance(ans, list)
and ans[0] != p_number_1
and ans[len(ans) - 1] != p_number_2
), "'ans' must been a list without the arguments"

# 'ans' contains not 'pNumber1' and 'pNumber2' !
return ans

# ----------------------------------------------------

def get_divisors(n):
"""
input: positive integer 'n' >= 1
returns all divisors of n (inclusive 1 and 'n')

>>> get_divisors(8)
[1, 2, 4, 8]
>>> get_divisors(824)
[1, 2, 4, 8, 103, 206, 412, 824]
>>> get_divisors(-1)
Traceback (most recent call last):
...
AssertionError: 'n' must been int and >= 1
>>> get_divisors("test")
Traceback (most recent call last):
...
AssertionError: 'n' must been int and >= 1
"""

# precondition
assert isinstance(n, int) and (n >= 1), "'n' must been int and >= 1"

ans = []  # will be returned.

for divisor in range(1, n + 1):
if n % divisor == 0:
ans.append(divisor)

# precondition
assert ans[0] == 1 and ans[len(ans) - 1] == n, "Error in function getDivisiors(...)"

return ans

# ----------------------------------------------------

def is_perfect_number(number):
"""
input: positive integer 'number' > 1
returns true if 'number' is a perfect number otherwise false.

>>> is_perfect_number(28)
True
>>> is_perfect_number(824)
False
>>> is_perfect_number(-1)
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and >= 1
>>> is_perfect_number("test")
Traceback (most recent call last):
...
AssertionError: 'number' must been an int and >= 1
"""

# precondition
assert isinstance(number, int) and (
number > 1
), "'number' must been an int and >= 1"

divisors = get_divisors(number)

# precondition
assert (
isinstance(divisors, list)
and (divisors[0] == 1)
and (divisors[len(divisors) - 1] == number)
), "Error in help-function getDivisiors(...)"

# summed all divisors up to 'number' (exclusive), hence [:-1]
return sum(divisors[:-1]) == number

# ------------------------------------------------------------

def simplify_fraction(numerator, denominator):
"""
input: two integer 'numerator' and 'denominator'
assumes: 'denominator' != 0
returns: a tuple with simplify numerator and denominator.

>>> simplify_fraction(10, 20)
(1, 2)
>>> simplify_fraction(10, -1)
(10, -1)
>>> simplify_fraction("test","test")
Traceback (most recent call last):
...
AssertionError: The arguments must been from type int and 'denominator' != 0
"""

# precondition
assert (
isinstance(numerator, int)
and isinstance(denominator, int)
and (denominator != 0)
), "The arguments must been from type int and 'denominator' != 0"

# build the greatest common divisor of numerator and denominator.
gcd_of_fraction = gcd_by_iterative(abs(numerator), abs(denominator))

# precondition
assert (
isinstance(gcd_of_fraction, int)
and (numerator % gcd_of_fraction == 0)
and (denominator % gcd_of_fraction == 0)
), "Error in function gcd_by_iterative(...,...)"

return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)

# -----------------------------------------------------------------

def factorial(n):
"""
input: positive integer 'n'
returns the factorial of 'n' (n!)

>>> factorial(0)
1
>>> factorial(20)
2432902008176640000
>>> factorial(-1)
Traceback (most recent call last):
...
AssertionError: 'n' must been a int and >= 0
>>> factorial("test")
Traceback (most recent call last):
...
AssertionError: 'n' must been a int and >= 0
"""

# precondition
assert isinstance(n, int) and (n >= 0), "'n' must been a int and >= 0"

ans = 1  # this will be return.

for factor in range(1, n + 1):
ans *= factor

return ans

# -------------------------------------------------------------------

def fib(n: int) -> int:
"""
input: positive integer 'n'
returns the n-th fibonacci term , indexing by 0

>>> fib(0)
1
>>> fib(5)
8
>>> fib(20)
10946
>>> fib(99)
354224848179261915075
>>> fib(-1)
Traceback (most recent call last):
...
AssertionError: 'n' must been an int and >= 0
>>> fib("test")
Traceback (most recent call last):
...
AssertionError: 'n' must been an int and >= 0
"""

# precondition
assert isinstance(n, int) and (n >= 0), "'n' must been an int and >= 0"

tmp = 0
fib1 = 1
ans = 1  # this will be return

for _ in range(n - 1):
tmp = ans
ans += fib1
fib1 = tmp

return ans

if __name__ == "__main__":
import doctest

doctest.testmod()
```