Python/project_euler/problem_46/sol1.py

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"""
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 × 12
15 = 7 + 2 × 22
21 = 3 + 2 × 32
25 = 7 + 2 × 32
27 = 19 + 2 × 22
33 = 31 + 2 × 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 typing import List
seive = [True] * 100001
i = 2
while i * i <= 100000:
if seive[i]:
for j in range(i * i, 100001, i):
seive[j] = False
i += 1
def is_prime(n: int) -> bool:
"""
Returns True if n is prime,
False otherwise, for 2 <= n <= 100000
>>> is_prime(87)
False
>>> is_prime(23)
True
>>> is_prime(25363)
False
"""
return seive[n]
odd_composites = [num for num in range(3, len(seive), 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
if __name__ == "__main__":
print(f"{compute_nums(1) = }")