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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.
isPrime(number)
getPrimeNumbers(N)
primeFactorization(number)
greatestPrimeFactor(number)
smallestPrimeFactor(number)
getPrime(n)
getPrimesBetween(pNumber1, pNumber2)
isEven(number)
isOdd(number)
gcd(number1, number2)  // greatest common divisor
kgV(number1, number2)  // least common multiple
getDivisors(number)    // all divisors of 'number' inclusive 1, number
isPerfectNumber(number)
NEW-FUNCTIONS
simplifyFraction(numerator, denominator)
factorial (n) // n!
fib (n) // calculate the n-th fibonacci term.
goldbach(number)  // Goldbach's assumption
from math import sqrt
def isPrime(number):
    input: positive integer 'number'
    returns true if 'number' is prime otherwise false.
    # 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 sieveEr(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.
    # 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
    beginList = [x for x in range(2, N + 1)]
    ans = []  # this list will be returns.
    # actual sieve of erathostenes
    for i in range(len(beginList)):
        for j in range(i + 1, len(beginList)):
            if (beginList[i] != 0) and (beginList[j] % beginList[i] == 0):
                beginList[j] = 0
    # filters actual prime numbers.
    ans = [x for x in beginList if x != 0]
    # precondition
    assert isinstance(ans, list), "'ans' must been from type list"
    return ans
# --------------------------------
def getPrimeNumbers(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(...)'
    # 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 isPrime(number):
            ans.append(number)
    # precondition
    assert isinstance(ans, list), "'ans' must been from type list"
    return ans
# -----------------------------------------
def primeFactorization(number):
    input: positive integer 'number'
    returns a list of the prime number factors of 'number'
    # 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 == 0 or number == 1:
        ans.append(number)
    # if 'number' not prime then builds the prime factorization of 'number'
    elif not isPrime(number):
        while quotient != 1:
            if isPrime(factor) and (quotient % factor == 0):
                ans.append(factor)
                quotient /= factor
            else:
                factor += 1
        ans.append(number)
    # precondition
    assert isinstance(ans, list), "'ans' must been from type list"
    return ans
# -----------------------------------------
def greatestPrimeFactor(number):
    input: positive integer 'number' >= 0
    returns the greatest prime number factor of 'number'
    # precondition
    assert isinstance(number, int) and (
        number >= 0
    ), "'number' bust been an int and >= 0"
    ans = 0
    # prime factorization of 'number'
    primeFactors = primeFactorization(number)
    ans = max(primeFactors)
    # precondition
    assert isinstance(ans, int), "'ans' must been from type int"
    return ans
# ----------------------------------------------
def smallestPrimeFactor(number):
    input: integer 'number' >= 0
    returns the smallest prime number factor of 'number'
    # precondition
    assert isinstance(number, int) and (
        number >= 0
    ), "'number' bust been an int and >= 0"
    ans = 0
    # prime factorization of 'number'
    primeFactors = primeFactorization(number)
    ans = min(primeFactors)
    # precondition
    assert isinstance(ans, int), "'ans' must been from type int"
    return ans
# ----------------------
def isEven(number):
    input: integer 'number'
    returns true if 'number' is even, otherwise false.
    # precondition
    assert isinstance(number, int), "'number' must been an int"
    assert isinstance(number % 2 == 0, bool), "compare bust been from type bool"
    return number % 2 == 0
# ------------------------
def isOdd(number):
    input: integer 'number'
    returns true if 'number' is odd, otherwise false.
    # precondition
    assert isinstance(number, int), "'number' must been an int"
    assert isinstance(number % 2 != 0, bool), "compare bust 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'
    # precondition
    assert (
        isinstance(number, int) and (number > 2) and isEven(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'
    primeNumbers = getPrimeNumbers(number)
    lenPN = len(primeNumbers)
    # run variable for while-loops.
    j = None
    # exit variable. for break up the loops
    loop = True
    while i < lenPN and loop:
        j = i + 1
        while j < lenPN and loop:
            if primeNumbers[i] + primeNumbers[j] == number:
                loop = False
                ans.append(primeNumbers[i])
                ans.append(primeNumbers[j])
        i += 1
    # precondition
    assert (
        isinstance(ans, list)
        and (len(ans) == 2)
        and (ans[0] + ans[1] == number)
        and isPrime(ans[0])
        and isPrime(ans[1])
    ), "'ans' must contains two primes. And sum of elements must been eq 'number'"
    return ans
# ----------------------------------------------
def gcd(number1, number2):
    Greatest common divisor
    input: two positive integer 'number1' and 'number2'
    returns the greatest common divisor of 'number1' and 'number2'
    # precondition
    assert (
        isinstance(number1, int)
        and isinstance(number2, int)
        and (number1 >= 0)
        and (number2 >= 0)
    ), "'number1' and 'number2' must been positive integer."
    rest = 0
    while number2 != 0:
        rest = number1 % number2
        number1 = number2
        number2 = rest
    # precondition
    assert isinstance(number1, int) and (
        number1 >= 0
    ), "'number' must been from type int and positive"
    return number1
# ----------------------------------------------------
def kgV(number1, number2):
    Least common multiple
    input: two positive integer 'number1' and 'number2'
    returns the least common multiple of 'number1' and 'number2'
    # 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'
        primeFac1 = primeFactorization(number1)
        primeFac2 = primeFactorization(number2)
    elif number1 == 1 or number2 == 1:
        primeFac1 = []
        primeFac2 = []
        ans = max(number1, number2)
    count1 = 0
    count2 = 0
    done = []  # captured numbers int both 'primeFac1' and 'primeFac2'
    # iterates through primeFac1
    for n in primeFac1:
        if n not in done:
            if n in primeFac2:
                count1 = primeFac1.count(n)
                count2 = primeFac2.count(n)
                for i in range(max(count1, count2)):
                    ans *= n
            else:
                count1 = primeFac1.count(n)
                for i in range(count1):
                    ans *= n
            done.append(n)
    # iterates through primeFac2
    for n in primeFac2:
        if n not in done:
            count2 = primeFac2.count(n)
            for i 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 getPrime(n):
    Gets the n-th prime number.
    input: positive integer 'n' >= 0
    returns the n-th prime number, beginning at index 0
    # 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 isPrime(ans):
            ans += 1
    # precondition
    assert isinstance(ans, int) and isPrime(
        ans
    ), "'ans' must been a prime number and from type int"
    return ans
# ---------------------------------------------------
def getPrimesBetween(pNumber1, pNumber2):
    input: prime numbers 'pNumber1' and 'pNumber2'
            pNumber1 < pNumber2
    returns a list of all prime numbers between 'pNumber1' (exclusive)
            and 'pNumber2' (exclusive)
    # precondition
    assert (
        isPrime(pNumber1) and isPrime(pNumber2) and (pNumber1 < pNumber2)
    ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"
    number = pNumber1 + 1  # jump to the next number
    ans = []  # this list will be returns.
    # if number is not prime then
    # fetch the next prime number.
    while not isPrime(number):
        number += 1
    while number < pNumber2:
        ans.append(number)
        number += 1
        # fetch the next prime number.
        while not isPrime(number):
            number += 1
    # precondition
    assert (
        isinstance(ans, list) and ans[0] != pNumber1 and ans[len(ans) - 1] != pNumber2
    ), "'ans' must been a list without the arguments"
    # 'ans' contains not 'pNumber1' and 'pNumber2' !
    return ans
# ----------------------------------------------------
def getDivisors(n):
    input: positive integer 'n' >= 1
    returns all divisors of n (inclusive 1 and 'n')
    # 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 isPerfectNumber(number):
    input: positive integer 'number' > 1
    returns true if 'number' is a perfect number otherwise false.
    # precondition
    assert isinstance(number, int) and (
        number > 1
    ), "'number' must been an int and >= 1"
    divisors = getDivisors(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 simplifyFraction(numerator, denominator):
    input: two integer 'numerator' and 'denominator'
    assumes: 'denominator' != 0
    returns: a tuple with simplify numerator and denominator.
    # 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.
    gcdOfFraction = gcd(abs(numerator), abs(denominator))
    # precondition
    assert (
        isinstance(gcdOfFraction, int)
        and (numerator % gcdOfFraction == 0)
        and (denominator % gcdOfFraction == 0)
    ), "Error in function gcd(...,...)"
    return (numerator // gcdOfFraction, denominator // gcdOfFraction)
# -----------------------------------------------------------------
def factorial(n):
    input: positive integer 'n'
    returns the factorial of 'n' (n!)
    # 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):
    input: positive integer 'n'
    returns the n-th fibonacci term , indexing by 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 i in range(n - 1):
        tmp = ans
        ans += fib1
        fib1 = tmp
    return ans
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