Hykilpikonna
158 lines
6.3 KiB
Python
158 lines
6.3 KiB
Python
"""CSC110 Fall 2021 Assignment 4, Part 2: Generating coprime numbers
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Instructions (READ THIS FIRST!)
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===============================
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Implement each of the functions in this file. As usual, do not change any function headers
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or preconditions. You do NOT need to add doctests.
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You may create additional helper functions to help break up your code into smaller parts.
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Copyright and Usage Information
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===============================
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This file is provided solely for the personal and private use of students
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taking CSC110 at the University of Toronto St. George campus. All forms of
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distribution of this code, whether as given or with any changes, are
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expressly prohibited. For more information on copyright for CSC110 materials,
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please consult our Course Syllabus.
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This file is Copyright (c) 2021 David Liu, Mario Badr, and Tom Fairgrieve.
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"""
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import math
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def coprime_to_2_and_3(n: int) -> list[int]:
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"""Return the natural numbers less than n that are coprime to both 2 and 3.
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The returned list is sorted.
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Preconditions:
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- n >= 6
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>>> coprime_to_2_and_3(20)
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[1, 5, 7, 11, 13, 17, 19]
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Implementation note: recall negative list indexing from Assignment 3.
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For all lists lst and integers i between 0 and len(lst) - 1 inclusive,
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lst[-i] == lst[len(lst) - i].
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"""
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nums_so_far = [1, 5]
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while nums_so_far[-2] + 6 < n:
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# Note: Write four assert statements here expressing the four loop invariants from the
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# assignment handout. These statements should be at the top of the loop body.
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# Loop Invariant 1: every number k in nums_so_far is coprime to 2 and coprime to 3.
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assert all(math.gcd(k, 2) == 1 and math.gcd(k, 3) == 1 for k in nums_so_far)
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# Loop Invariant 2: for all natural numbers i between 0 and len(nums_so_far) - 3 inclusive,
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# nums_so_far[i] + 6 == nums_so_far[i + 2].
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assert all(nums_so_far[i] + 6 == nums_so_far[i + 2] for i in range(len(nums_so_far) - 2))
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# Loop Invariant 3: for all natural numbers i between 0 and len(nums_so_far) - 2 inclusive,
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# nums_so_far[i] < nums_so_far[i + 1] (this means that nums_so_far is always sorted).
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assert all(nums_so_far[i] < nums_so_far[i + 1] for i in range(len(nums_so_far) - 1))
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# Loop Invariant 4: for all natural numbers k between 0 and nums_so_far[-1] inclusive,
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# if k is coprime to 2 and coprime to 3, then k in nums_so_far.
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assert all(k in nums_so_far for k in range(nums_so_far[-1] + 1)
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if math.gcd(k, 2) == 1 and math.gcd(k, 3) == 1)
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next_number = nums_so_far[-2] + 6
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list.append(nums_so_far, next_number)
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return nums_so_far
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def coprime_to_all(primes: set[int], n: int) -> list[int]:
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"""Return the positive integers less than n that are coprime to every number in primes.
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The returned list is sorted.
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Pay attention to the preconditions, as they are designed to help simplify your work for this
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question.
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Preconditions:
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- primes != set()
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- every element of primes is prime
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- n >= math.prod(primes)
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>>> coprime_to_all({2, 3}, 20)
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[1, 5, 7, 11, 13, 17, 19]
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>>> coprime_to_all({2, 3, 7}, 50)
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[1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47]
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Implementation notes:
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- You MUST use the provided helper function starting_coprime_numbers in your implementation,
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and may NOT modify it (even though it is not as efficient as it could be!!).
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- You will find the math.prod function useful.
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"""
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increment = math.prod(primes)
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nums_so_far = starting_coprime_numbers(primes)
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period = len(nums_so_far)
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while nums_so_far[-period] + increment < n:
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# Note: Write four assert statements here expressing the four loop invariants from the
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# assignment handout. These statements should be at the top of the loop body.
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# Loop Invariant 1: every number k in nums_so_far is coprime to every prime
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assert all(all(math.gcd(p, k) == 1 for p in primes) for k in nums_so_far)
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# Loop Invariant 2: for all natural numbers i between 0 and
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# len(nums_so_far) - period - 1 inclusive,
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# nums_so_far[i] + increment == nums_so_far[i + period].
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assert all(nums_so_far[i] + increment == nums_so_far[i + period]
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for i in range(len(nums_so_far) - period))
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# Loop Invariant 3: for all natural numbers i between 0 and len(nums_so_far) - 2 inclusive,
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# nums_so_far[i] < nums_so_far[i + 1] (this means that nums_so_far is always sorted).
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assert all(nums_so_far[i] < nums_so_far[i + 1] for i in range(len(nums_so_far) - 2 + 1))
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# Loop Invariant 4: for all natural numbers k between 0 and nums_so_far[-1] inclusive,
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# if k is coprime to all primes, then k in nums_so_far.
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assert all(k in nums_so_far for k in range(nums_so_far[-1] + 1)
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if all(math.gcd(p, k) == 1 for p in primes))
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next_number = nums_so_far[-period] + increment
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list.append(nums_so_far, next_number)
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return nums_so_far
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def starting_coprime_numbers(primes: set[int]) -> list[int]:
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"""Return the numbers up to the product of the given primes that are coprime to all of them.
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Note: the length of the returned list is is exactly equal to phi(math.prod(primes)), where
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phi is the Euler totient function.
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Preconditions:
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- primes != set()
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- every element of primes is prime
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>>> starting_coprime_numbers({2, 3})
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[1, 5]
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>>> starting_coprime_numbers({3, 11})
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[1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32]
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"""
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nums_so_far = []
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m = math.prod(primes)
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for k in range(1, m):
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is_coprime = True
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for p in primes:
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if k % p == 0:
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is_coprime = False
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if is_coprime:
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list.append(nums_so_far, k)
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return nums_so_far
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if __name__ == '__main__':
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# When you are ready to check your work with python_ta, uncomment the following lines.
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# (Delete the "#" and space before each line.)
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# IMPORTANT: keep this code indented inside the "if __name__ == '__main__'" block
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# Leave this code uncommented when you submit your files.
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import python_ta
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python_ta.check_all(config={
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'extra-imports': ['python_ta.contracts', 'math'],
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'max-line-length': 100,
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'disable': ['R1705', 'C0200']
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})
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import doctest
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doctest.testmod()
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