# -*- coding: utf-8 -*-
"""
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:
isPrime(number)
sieveEr(N)
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
"""
def isPrime(number):
"""
input: positive integer 'number'
returns true if 'number' is prime otherwise false.
"""
import math # for function sqrt
# 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(math.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: conatins all natural numbers from 2 upt 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'
"""
import math # for function sqrt
# 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
else:
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.
i = 0
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])
j += 1
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' (exclusiv)
and 'pNumber2' (exclusiv)
"""
# 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"
from math import sqrt
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