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2104fa7aeb
* fixes #5434 * fixes broken solution * removes assert * removes assert * Apply suggestions from code review Co-authored-by: John Law <johnlaw.po@gmail.com> * Update project_euler/problem_003/sol1.py Co-authored-by: John Law <johnlaw.po@gmail.com>
117 lines
2.7 KiB
Python
117 lines
2.7 KiB
Python
"""
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Problem 46: https://projecteuler.net/problem=46
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It was proposed by Christian Goldbach that every odd composite number can be
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written as the sum of a prime and twice a square.
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9 = 7 + 2 × 12
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15 = 7 + 2 × 22
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21 = 3 + 2 × 32
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25 = 7 + 2 × 32
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27 = 19 + 2 × 22
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33 = 31 + 2 × 12
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It turns out that the conjecture was false.
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What is the smallest odd composite that cannot be written as the sum of a
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prime and twice a square?
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"""
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from __future__ import annotations
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import math
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def is_prime(number: int) -> bool:
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"""Checks to see if a number is a prime in O(sqrt(n)).
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A number is prime if it has exactly two factors: 1 and itself.
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>>> is_prime(0)
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False
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>>> is_prime(1)
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False
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>>> is_prime(2)
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True
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>>> is_prime(3)
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True
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>>> is_prime(27)
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False
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>>> is_prime(87)
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False
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>>> is_prime(563)
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True
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>>> is_prime(2999)
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True
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>>> is_prime(67483)
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False
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"""
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if 1 < number < 4:
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# 2 and 3 are primes
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return True
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elif number < 2 or number % 2 == 0 or number % 3 == 0:
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# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
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return False
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# All primes number are in format of 6k +/- 1
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for i in range(5, int(math.sqrt(number) + 1), 6):
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if number % i == 0 or number % (i + 2) == 0:
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return False
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return True
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odd_composites = [num for num in range(3, 100001, 2) if not is_prime(num)]
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def compute_nums(n: int) -> list[int]:
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"""
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Returns a list of first n odd composite numbers which do
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not follow the conjecture.
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>>> compute_nums(1)
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[5777]
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>>> compute_nums(2)
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[5777, 5993]
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>>> compute_nums(0)
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Traceback (most recent call last):
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...
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ValueError: n must be >= 0
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>>> compute_nums("a")
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Traceback (most recent call last):
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...
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ValueError: n must be an integer
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>>> compute_nums(1.1)
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Traceback (most recent call last):
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...
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ValueError: n must be an integer
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"""
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if not isinstance(n, int):
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raise ValueError("n must be an integer")
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if n <= 0:
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raise ValueError("n must be >= 0")
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list_nums = []
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for num in range(len(odd_composites)):
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i = 0
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while 2 * i * i <= odd_composites[num]:
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rem = odd_composites[num] - 2 * i * i
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if is_prime(rem):
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break
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i += 1
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else:
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list_nums.append(odd_composites[num])
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if len(list_nums) == n:
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return list_nums
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return []
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def solution() -> int:
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"""Return the solution to the problem"""
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return compute_nums(1)[0]
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if __name__ == "__main__":
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print(f"{solution() = }")
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