mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-12-18 17:20:16 +00:00
4700297b3e
* Enable ruff RUF002 rule * Fix --------- Co-authored-by: Christian Clauss <cclauss@me.com>
117 lines
2.7 KiB
Python
117 lines
2.7 KiB
Python
"""
|
|
Problem 46: https://projecteuler.net/problem=46
|
|
|
|
It was proposed by Christian Goldbach that every odd composite number can be
|
|
written as the sum of a prime and twice a square.
|
|
|
|
9 = 7 + 2 x 12
|
|
15 = 7 + 2 x 22
|
|
21 = 3 + 2 x 32
|
|
25 = 7 + 2 x 32
|
|
27 = 19 + 2 x 22
|
|
33 = 31 + 2 x 12
|
|
|
|
It turns out that the conjecture was false.
|
|
|
|
What is the smallest odd composite that cannot be written as the sum of a
|
|
prime and twice a square?
|
|
"""
|
|
|
|
from __future__ import annotations
|
|
|
|
import math
|
|
|
|
|
|
def is_prime(number: int) -> bool:
|
|
"""Checks to see if a number is a prime in O(sqrt(n)).
|
|
|
|
A number is prime if it has exactly two factors: 1 and itself.
|
|
|
|
>>> is_prime(0)
|
|
False
|
|
>>> is_prime(1)
|
|
False
|
|
>>> is_prime(2)
|
|
True
|
|
>>> is_prime(3)
|
|
True
|
|
>>> is_prime(27)
|
|
False
|
|
>>> is_prime(87)
|
|
False
|
|
>>> is_prime(563)
|
|
True
|
|
>>> is_prime(2999)
|
|
True
|
|
>>> is_prime(67483)
|
|
False
|
|
"""
|
|
|
|
if 1 < number < 4:
|
|
# 2 and 3 are primes
|
|
return True
|
|
elif number < 2 or number % 2 == 0 or number % 3 == 0:
|
|
# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
|
|
return False
|
|
|
|
# All primes number are in format of 6k +/- 1
|
|
for i in range(5, int(math.sqrt(number) + 1), 6):
|
|
if number % i == 0 or number % (i + 2) == 0:
|
|
return False
|
|
return True
|
|
|
|
|
|
odd_composites = [num for num in range(3, 100001, 2) if not is_prime(num)]
|
|
|
|
|
|
def compute_nums(n: int) -> list[int]:
|
|
"""
|
|
Returns a list of first n odd composite numbers which do
|
|
not follow the conjecture.
|
|
>>> compute_nums(1)
|
|
[5777]
|
|
>>> compute_nums(2)
|
|
[5777, 5993]
|
|
>>> compute_nums(0)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n must be >= 0
|
|
>>> compute_nums("a")
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n must be an integer
|
|
>>> compute_nums(1.1)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n must be an integer
|
|
|
|
"""
|
|
if not isinstance(n, int):
|
|
raise ValueError("n must be an integer")
|
|
if n <= 0:
|
|
raise ValueError("n must be >= 0")
|
|
|
|
list_nums = []
|
|
for num in range(len(odd_composites)):
|
|
i = 0
|
|
while 2 * i * i <= odd_composites[num]:
|
|
rem = odd_composites[num] - 2 * i * i
|
|
if is_prime(rem):
|
|
break
|
|
i += 1
|
|
else:
|
|
list_nums.append(odd_composites[num])
|
|
if len(list_nums) == n:
|
|
return list_nums
|
|
|
|
return []
|
|
|
|
|
|
def solution() -> int:
|
|
"""Return the solution to the problem"""
|
|
return compute_nums(1)[0]
|
|
|
|
|
|
if __name__ == "__main__":
|
|
print(f"{solution() = }")
|