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842 lines
20 KiB
Python
842 lines
20 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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is_prime(number)
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sieve_er(N)
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get_prime_numbers(N)
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prime_factorization(number)
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greatest_prime_factor(number)
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smallest_prime_factor(number)
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get_prime(n)
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get_primes_between(pNumber1, pNumber2)
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----
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is_even(number)
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is_odd(number)
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kg_v(number1, number2) // least common multiple
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get_divisors(number) // all divisors of 'number' inclusive 1, number
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is_perfect_number(number)
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NEW-FUNCTIONS
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simplify_fraction(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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from maths.greatest_common_divisor import gcd_by_iterative
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def is_prime(number: int) -> bool:
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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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>>> is_prime(3)
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True
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>>> is_prime(10)
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False
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>>> is_prime(97)
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True
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>>> is_prime(9991)
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False
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>>> is_prime(-1)
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and positive
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>>> is_prime("test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and positive
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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 sieve_er(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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>>> sieve_er(8)
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[2, 3, 5, 7]
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>>> sieve_er(-1)
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Traceback (most recent call last):
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...
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AssertionError: 'N' must been an int and > 2
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>>> sieve_er("test")
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Traceback (most recent call last):
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...
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AssertionError: 'N' must been an int and > 2
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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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begin_list = list(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(begin_list)):
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for j in range(i + 1, len(begin_list)):
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if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
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begin_list[j] = 0
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# filters actual prime numbers.
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ans = [x for x in begin_list 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 get_prime_numbers(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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>>> get_prime_numbers(8)
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[2, 3, 5, 7]
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>>> get_prime_numbers(-1)
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Traceback (most recent call last):
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...
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AssertionError: 'N' must been an int and > 2
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>>> get_prime_numbers("test")
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Traceback (most recent call last):
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...
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AssertionError: 'N' must been an int and > 2
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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 is_prime(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 prime_factorization(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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>>> prime_factorization(0)
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[0]
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>>> prime_factorization(8)
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[2, 2, 2]
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>>> prime_factorization(287)
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[7, 41]
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>>> prime_factorization(-1)
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and >= 0
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>>> prime_factorization("test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and >= 0
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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 in {0, 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 is_prime(number):
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while quotient != 1:
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if is_prime(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 greatest_prime_factor(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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>>> greatest_prime_factor(0)
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0
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>>> greatest_prime_factor(8)
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2
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>>> greatest_prime_factor(287)
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41
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>>> greatest_prime_factor(-1)
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and >= 0
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>>> greatest_prime_factor("test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and >= 0
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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 >= 0"
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ans = 0
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# prime factorization of 'number'
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prime_factors = prime_factorization(number)
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ans = max(prime_factors)
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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 smallest_prime_factor(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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>>> smallest_prime_factor(0)
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0
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>>> smallest_prime_factor(8)
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2
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>>> smallest_prime_factor(287)
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7
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>>> smallest_prime_factor(-1)
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and >= 0
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>>> smallest_prime_factor("test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and >= 0
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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 >= 0"
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ans = 0
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# prime factorization of 'number'
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prime_factors = prime_factorization(number)
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ans = min(prime_factors)
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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 is_even(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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>>> is_even(0)
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True
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>>> is_even(8)
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True
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>>> is_even(287)
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False
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>>> is_even(-1)
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False
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>>> is_even("test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int
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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 must been from type bool"
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return number % 2 == 0
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# ------------------------
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def is_odd(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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>>> is_odd(0)
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False
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>>> is_odd(8)
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False
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>>> is_odd(287)
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True
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>>> is_odd(-1)
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True
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>>> is_odd("test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int
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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 must 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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>>> goldbach(8)
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[3, 5]
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>>> goldbach(824)
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[3, 821]
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>>> goldbach(0)
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int, even and > 2
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>>> goldbach(-1)
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int, even and > 2
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>>> goldbach("test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int, even and > 2
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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 is_even(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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prime_numbers = get_prime_numbers(number)
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len_pn = len(prime_numbers)
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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 < len_pn and loop:
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j = i + 1
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while j < len_pn and loop:
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if prime_numbers[i] + prime_numbers[j] == number:
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loop = False
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ans.append(prime_numbers[i])
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ans.append(prime_numbers[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 is_prime(ans[0])
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and is_prime(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 kg_v(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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>>> kg_v(8,10)
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40
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>>> kg_v(824,67)
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55208
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>>> kg_v(1, 10)
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10
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>>> kg_v(0)
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Traceback (most recent call last):
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...
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TypeError: kg_v() missing 1 required positional argument: 'number2'
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>>> kg_v(10,-1)
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Traceback (most recent call last):
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...
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AssertionError: 'number1' and 'number2' must been positive integer.
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>>> kg_v("test","test2")
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Traceback (most recent call last):
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...
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AssertionError: 'number1' and 'number2' must been positive integer.
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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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prime_fac_1 = prime_factorization(number1)
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prime_fac_2 = prime_factorization(number2)
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elif number1 == 1 or number2 == 1:
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prime_fac_1 = []
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prime_fac_2 = []
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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 prime_fac_1:
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if n not in done:
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if n in prime_fac_2:
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count1 = prime_fac_1.count(n)
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count2 = prime_fac_2.count(n)
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for _ in range(max(count1, count2)):
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ans *= n
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else:
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count1 = prime_fac_1.count(n)
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for _ 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 prime_fac_2:
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if n not in done:
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count2 = prime_fac_2.count(n)
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for _ 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 get_prime(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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>>> get_prime(0)
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2
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>>> get_prime(8)
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23
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>>> get_prime(824)
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6337
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>>> get_prime(-1)
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been a positive int
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>>> get_prime("test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been a positive int
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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 is_prime(ans):
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ans += 1
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# precondition
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assert isinstance(ans, int) and is_prime(
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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 get_primes_between(p_number_1, p_number_2):
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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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>>> get_primes_between(3, 67)
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[5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61]
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>>> get_primes_between(0)
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Traceback (most recent call last):
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...
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TypeError: get_primes_between() missing 1 required positional argument: 'p_number_2'
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>>> get_primes_between(0, 1)
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Traceback (most recent call last):
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...
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AssertionError: The arguments must been prime numbers and 'pNumber1' < 'pNumber2'
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>>> get_primes_between(-1, 3)
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and positive
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>>> get_primes_between("test","test")
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Traceback (most recent call last):
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...
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AssertionError: 'number' must been an int and positive
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"""
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# precondition
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assert (
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is_prime(p_number_1) and is_prime(p_number_2) and (p_number_1 < p_number_2)
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), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"
|
|
|
|
number = p_number_1 + 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 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()
|