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645 lines
14 KiB
Python
645 lines
14 KiB
Python
"""
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Created on Thu Oct 5 16:44:23 2017
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@author: Christian Bender
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This Python library contains some useful functions to deal with
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prime numbers and whole numbers.
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Overview:
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isPrime(number)
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sieveEr(N)
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getPrimeNumbers(N)
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primeFactorization(number)
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greatestPrimeFactor(number)
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smallestPrimeFactor(number)
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getPrime(n)
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getPrimesBetween(pNumber1, pNumber2)
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----
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isEven(number)
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isOdd(number)
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gcd(number1, number2) // greatest common divisor
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kgV(number1, number2) // least common multiple
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getDivisors(number) // all divisors of 'number' inclusive 1, number
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isPerfectNumber(number)
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NEW-FUNCTIONS
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simplifyFraction(numerator, denominator)
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factorial (n) // n!
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fib (n) // calculate the n-th fibonacci term.
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-----
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goldbach(number) // Goldbach's assumption
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"""
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from math import sqrt
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def isPrime(number):
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"""
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input: positive integer 'number'
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returns true if 'number' is prime otherwise false.
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"""
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# precondition
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assert isinstance(number, int) and (
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number >= 0
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), "'number' must been an int and positive"
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status = True
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# 0 and 1 are none primes.
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if number <= 1:
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status = False
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for divisor in range(2, int(round(sqrt(number))) + 1):
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# if 'number' divisible by 'divisor' then sets 'status'
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# of false and break up the loop.
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if number % divisor == 0:
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status = False
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break
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# precondition
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assert isinstance(status, bool), "'status' must been from type bool"
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return status
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# ------------------------------------------
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def sieveEr(N):
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"""
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input: positive integer 'N' > 2
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returns a list of prime numbers from 2 up to N.
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This function implements the algorithm called
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sieve of erathostenes.
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"""
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# precondition
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assert isinstance(N, int) and (N > 2), "'N' must been an int and > 2"
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# beginList: contains all natural numbers from 2 up to N
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beginList = [x for x in range(2, N + 1)]
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ans = [] # this list will be returns.
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# actual sieve of erathostenes
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for i in range(len(beginList)):
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for j in range(i + 1, len(beginList)):
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if (beginList[i] != 0) and (beginList[j] % beginList[i] == 0):
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beginList[j] = 0
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# filters actual prime numbers.
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ans = [x for x in beginList if x != 0]
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# precondition
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assert isinstance(ans, list), "'ans' must been from type list"
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return ans
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# --------------------------------
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def getPrimeNumbers(N):
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"""
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input: positive integer 'N' > 2
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returns a list of prime numbers from 2 up to N (inclusive)
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This function is more efficient as function 'sieveEr(...)'
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"""
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# precondition
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assert isinstance(N, int) and (N > 2), "'N' must been an int and > 2"
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ans = []
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# iterates over all numbers between 2 up to N+1
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# if a number is prime then appends to list 'ans'
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for number in range(2, N + 1):
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if isPrime(number):
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ans.append(number)
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# precondition
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assert isinstance(ans, list), "'ans' must been from type list"
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return ans
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# -----------------------------------------
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def primeFactorization(number):
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"""
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input: positive integer 'number'
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returns a list of the prime number factors of 'number'
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"""
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# precondition
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assert isinstance(number, int) and number >= 0, "'number' must been an int and >= 0"
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ans = [] # this list will be returns of the function.
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# potential prime number factors.
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factor = 2
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quotient = number
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if number == 0 or number == 1:
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ans.append(number)
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# if 'number' not prime then builds the prime factorization of 'number'
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elif not isPrime(number):
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while quotient != 1:
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if isPrime(factor) and (quotient % factor == 0):
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ans.append(factor)
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quotient /= factor
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else:
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factor += 1
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else:
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ans.append(number)
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# precondition
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assert isinstance(ans, list), "'ans' must been from type list"
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return ans
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# -----------------------------------------
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def greatestPrimeFactor(number):
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"""
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input: positive integer 'number' >= 0
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returns the greatest prime number factor of 'number'
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"""
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# precondition
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assert isinstance(number, int) and (
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number >= 0
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), "'number' bust been an int and >= 0"
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ans = 0
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# prime factorization of 'number'
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primeFactors = primeFactorization(number)
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ans = max(primeFactors)
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# precondition
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assert isinstance(ans, int), "'ans' must been from type int"
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return ans
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# ----------------------------------------------
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def smallestPrimeFactor(number):
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"""
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input: integer 'number' >= 0
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returns the smallest prime number factor of 'number'
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"""
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# precondition
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assert isinstance(number, int) and (
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number >= 0
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), "'number' bust been an int and >= 0"
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ans = 0
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# prime factorization of 'number'
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primeFactors = primeFactorization(number)
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ans = min(primeFactors)
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# precondition
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assert isinstance(ans, int), "'ans' must been from type int"
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return ans
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# ----------------------
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def isEven(number):
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"""
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input: integer 'number'
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returns true if 'number' is even, otherwise false.
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"""
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# precondition
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assert isinstance(number, int), "'number' must been an int"
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assert isinstance(number % 2 == 0, bool), "compare bust been from type bool"
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return number % 2 == 0
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# ------------------------
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def isOdd(number):
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"""
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input: integer 'number'
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returns true if 'number' is odd, otherwise false.
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"""
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# precondition
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assert isinstance(number, int), "'number' must been an int"
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assert isinstance(number % 2 != 0, bool), "compare bust been from type bool"
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return number % 2 != 0
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# ------------------------
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def goldbach(number):
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"""
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Goldbach's assumption
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input: a even positive integer 'number' > 2
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returns a list of two prime numbers whose sum is equal to 'number'
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"""
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# precondition
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assert (
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isinstance(number, int) and (number > 2) and isEven(number)
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), "'number' must been an int, even and > 2"
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ans = [] # this list will returned
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# creates a list of prime numbers between 2 up to 'number'
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primeNumbers = getPrimeNumbers(number)
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lenPN = len(primeNumbers)
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# run variable for while-loops.
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i = 0
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j = None
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# exit variable. for break up the loops
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loop = True
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while i < lenPN and loop:
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j = i + 1
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while j < lenPN and loop:
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if primeNumbers[i] + primeNumbers[j] == number:
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loop = False
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ans.append(primeNumbers[i])
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ans.append(primeNumbers[j])
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j += 1
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i += 1
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# precondition
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assert (
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isinstance(ans, list)
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and (len(ans) == 2)
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and (ans[0] + ans[1] == number)
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and isPrime(ans[0])
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and isPrime(ans[1])
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), "'ans' must contains two primes. And sum of elements must been eq 'number'"
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return ans
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# ----------------------------------------------
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def gcd(number1, number2):
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"""
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Greatest common divisor
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input: two positive integer 'number1' and 'number2'
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returns the greatest common divisor of 'number1' and 'number2'
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"""
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# precondition
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assert (
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isinstance(number1, int)
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and isinstance(number2, int)
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and (number1 >= 0)
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and (number2 >= 0)
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), "'number1' and 'number2' must been positive integer."
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rest = 0
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while number2 != 0:
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rest = number1 % number2
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number1 = number2
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number2 = rest
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# precondition
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assert isinstance(number1, int) and (
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number1 >= 0
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), "'number' must been from type int and positive"
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return number1
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# ----------------------------------------------------
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def kgV(number1, number2):
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"""
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Least common multiple
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input: two positive integer 'number1' and 'number2'
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returns the least common multiple of 'number1' and 'number2'
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"""
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# precondition
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assert (
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isinstance(number1, int)
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and isinstance(number2, int)
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and (number1 >= 1)
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and (number2 >= 1)
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), "'number1' and 'number2' must been positive integer."
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ans = 1 # actual answer that will be return.
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# for kgV (x,1)
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if number1 > 1 and number2 > 1:
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# builds the prime factorization of 'number1' and 'number2'
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primeFac1 = primeFactorization(number1)
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primeFac2 = primeFactorization(number2)
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elif number1 == 1 or number2 == 1:
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primeFac1 = []
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primeFac2 = []
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ans = max(number1, number2)
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count1 = 0
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count2 = 0
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done = [] # captured numbers int both 'primeFac1' and 'primeFac2'
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# iterates through primeFac1
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for n in primeFac1:
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if n not in done:
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if n in primeFac2:
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count1 = primeFac1.count(n)
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count2 = primeFac2.count(n)
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for i in range(max(count1, count2)):
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ans *= n
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else:
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count1 = primeFac1.count(n)
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for i in range(count1):
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ans *= n
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done.append(n)
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# iterates through primeFac2
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for n in primeFac2:
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if n not in done:
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count2 = primeFac2.count(n)
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for i in range(count2):
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ans *= n
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done.append(n)
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# precondition
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assert isinstance(ans, int) and (
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ans >= 0
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), "'ans' must been from type int and positive"
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return ans
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# ----------------------------------
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def getPrime(n):
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"""
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Gets the n-th prime number.
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input: positive integer 'n' >= 0
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returns the n-th prime number, beginning at index 0
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"""
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# precondition
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assert isinstance(n, int) and (n >= 0), "'number' must been a positive int"
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index = 0
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ans = 2 # this variable holds the answer
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while index < n:
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index += 1
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ans += 1 # counts to the next number
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# if ans not prime then
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# runs to the next prime number.
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while not isPrime(ans):
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ans += 1
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# precondition
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assert isinstance(ans, int) and isPrime(
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ans
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), "'ans' must been a prime number and from type int"
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return ans
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# ---------------------------------------------------
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def getPrimesBetween(pNumber1, pNumber2):
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"""
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input: prime numbers 'pNumber1' and 'pNumber2'
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pNumber1 < pNumber2
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returns a list of all prime numbers between 'pNumber1' (exclusive)
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and 'pNumber2' (exclusive)
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"""
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# precondition
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assert (
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isPrime(pNumber1) and isPrime(pNumber2) and (pNumber1 < pNumber2)
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), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"
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number = pNumber1 + 1 # jump to the next number
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ans = [] # this list will be returns.
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# if number is not prime then
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# fetch the next prime number.
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while not isPrime(number):
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number += 1
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while number < pNumber2:
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ans.append(number)
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number += 1
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# fetch the next prime number.
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while not isPrime(number):
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number += 1
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# precondition
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assert (
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isinstance(ans, list) and ans[0] != pNumber1 and ans[len(ans) - 1] != pNumber2
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), "'ans' must been a list without the arguments"
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# 'ans' contains not 'pNumber1' and 'pNumber2' !
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return ans
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# ----------------------------------------------------
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def getDivisors(n):
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"""
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input: positive integer 'n' >= 1
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returns all divisors of n (inclusive 1 and 'n')
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"""
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# precondition
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assert isinstance(n, int) and (n >= 1), "'n' must been int and >= 1"
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ans = [] # will be returned.
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for divisor in range(1, n + 1):
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if n % divisor == 0:
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ans.append(divisor)
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# precondition
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assert ans[0] == 1 and ans[len(ans) - 1] == n, "Error in function getDivisiors(...)"
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return ans
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# ----------------------------------------------------
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def isPerfectNumber(number):
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"""
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input: positive integer 'number' > 1
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returns true if 'number' is a perfect number otherwise false.
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"""
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# precondition
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assert isinstance(number, int) and (
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number > 1
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), "'number' must been an int and >= 1"
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divisors = getDivisors(number)
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# precondition
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assert (
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isinstance(divisors, list)
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and (divisors[0] == 1)
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and (divisors[len(divisors) - 1] == number)
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), "Error in help-function getDivisiors(...)"
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# summed all divisors up to 'number' (exclusive), hence [:-1]
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return sum(divisors[:-1]) == number
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# ------------------------------------------------------------
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def simplifyFraction(numerator, denominator):
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"""
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input: two integer 'numerator' and 'denominator'
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assumes: 'denominator' != 0
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returns: a tuple with simplify numerator and denominator.
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"""
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# precondition
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assert (
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isinstance(numerator, int)
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and isinstance(denominator, int)
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and (denominator != 0)
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), "The arguments must been from type int and 'denominator' != 0"
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# build the greatest common divisor of numerator and denominator.
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gcdOfFraction = gcd(abs(numerator), abs(denominator))
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# precondition
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assert (
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isinstance(gcdOfFraction, int)
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and (numerator % gcdOfFraction == 0)
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and (denominator % gcdOfFraction == 0)
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), "Error in function gcd(...,...)"
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return (numerator // gcdOfFraction, denominator // gcdOfFraction)
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# -----------------------------------------------------------------
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def factorial(n):
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"""
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input: positive integer 'n'
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returns the factorial of 'n' (n!)
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"""
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# precondition
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assert isinstance(n, int) and (n >= 0), "'n' must been a int and >= 0"
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ans = 1 # this will be return.
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for factor in range(1, n + 1):
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ans *= factor
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return ans
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# -------------------------------------------------------------------
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def fib(n):
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"""
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input: positive integer 'n'
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returns the n-th fibonacci term , indexing by 0
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"""
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# precondition
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assert isinstance(n, int) and (n >= 0), "'n' must been an int and >= 0"
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tmp = 0
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fib1 = 1
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ans = 1 # this will be return
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for i in range(n - 1):
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tmp = ans
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ans += fib1
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fib1 = tmp
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return ans
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