"""
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 * 1^2
15 = 7 + 2 * 2^2
21 = 3 + 2 * 3^2
25 = 7 + 2 * 3^2
27 = 19 + 2 * 2^2
33 = 31 + 2 * 1^2

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?
"""

import math
import time


def timer(func):
    def wrapper(*args, **kwargs):
        start_time = time.time()
        result = func(*args, **kwargs)
        end_time = time.time()
        print(f"{func.__name__} took {end_time - start_time:.6f} seconds")
        return result

    return wrapper


def sieve_of_eratosthenes(limit: int = 5000) -> list[bool]:
    """生成一个布尔数组，is_prime[i]为True表示i是质数"""
    is_prime = [True] * (limit + 1)
    is_prime[0:2] = [False, False]

    for i in range(2, int(limit**0.5) + 1):
        if is_prime[i]:
            for j in range(i * i, limit + 1, i):
                is_prime[j] = False
    return is_prime


def optimized_find_counterexamples(limit: int = 10000) -> list[int]:
    """
    优化版本，使用集合提高查找效率
    """
    # 生成质数表
    is_prime = sieve_of_eratosthenes(limit)
    primes = [i for i in range(limit + 1) if is_prime[i]]

    # 预计算所有奇合数
    odd_composites = set(n for n in range(3, limit + 1, 2) if not is_prime[n])

    # 预计算2k²的值
    max_k = int(math.sqrt(limit / 2)) + 1
    two_k_squared = [2 * k * k for k in range(1, max_k + 1)]

    # 计算所有可以表示的奇合数
    representable_odd_composites = set()
    for p in primes:
        if p > limit:
            break
        for tks in two_k_squared:
            n = p + tks
            if n > limit:
                break
            if n in odd_composites:
                representable_odd_composites.add(n)

    # 找出不能表示的奇合数
    return sorted(odd_composites - representable_odd_composites)


@timer
def main(limit: int = 20000) -> None:
    res = optimized_find_counterexamples(limit)
    for r in res:
        print(r)


if __name__ == "__main__":
    main()