82 lines
2.2 KiB
Python
82 lines
2.2 KiB
Python
"""
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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.
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9 = 7 + 2 * 1^2
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15 = 7 + 2 * 2^2
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21 = 3 + 2 * 3^2
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25 = 7 + 2 * 3^2
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27 = 19 + 2 * 2^2
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33 = 31 + 2 * 1^2
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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 prime and twice a square?
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"""
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import math
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import time
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def timer(func):
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def wrapper(*args, **kwargs):
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start_time = time.time()
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result = func(*args, **kwargs)
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end_time = time.time()
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print(f"{func.__name__} took {end_time - start_time:.6f} seconds")
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return result
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return wrapper
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def sieve_of_eratosthenes(limit: int = 5000) -> list[bool]:
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"""生成一个布尔数组，is_prime[i]为True表示i是质数"""
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is_prime = [True] * (limit + 1)
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is_prime[0:2] = [False, False]
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for i in range(2, int(limit**0.5) + 1):
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if is_prime[i]:
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for j in range(i * i, limit + 1, i):
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is_prime[j] = False
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return is_prime
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def optimized_find_counterexamples(limit: int = 10000) -> list[int]:
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"""
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优化版本，使用集合提高查找效率
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"""
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# 生成质数表
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is_prime = sieve_of_eratosthenes(limit)
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primes = [i for i in range(limit + 1) if is_prime[i]]
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# 预计算所有奇合数
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odd_composites = set(n for n in range(3, limit + 1, 2) if not is_prime[n])
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# 预计算2k²的值
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max_k = int(math.sqrt(limit / 2)) + 1
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two_k_squared = [2 * k * k for k in range(1, max_k + 1)]
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# 计算所有可以表示的奇合数
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representable_odd_composites = set()
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for p in primes:
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if p > limit:
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break
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for tks in two_k_squared:
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n = p + tks
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if n > limit:
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break
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if n in odd_composites:
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representable_odd_composites.add(n)
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# 找出不能表示的奇合数
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return sorted(odd_composites - representable_odd_composites)
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@timer
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def main(limit: int = 20000) -> None:
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res = optimized_find_counterexamples(limit)
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for r in res:
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print(r)
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if __name__ == "__main__":
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main()
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