106 lines
3.3 KiB
Python
106 lines
3.3 KiB
Python
"""
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By replacing the 1st digit of the 2-digit number *3,
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it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.
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By replacing the 3rd and 4th digits of 56**3 with the same digit,
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this 5-digit number is the first example having seven primes among the ten generated numbers,
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yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993.
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Consequently 56003, being the first member of this family, is the smallest prime with this property.
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Find the smallest prime which,
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by replacing part of the number (not necessarily adjacent digits) with the same digit,
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is part of an eight prime value family.
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-----
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*.000*.、*.111*.、*.222*.、*.333*.、*.444*.、*.555*.、*.666*.、*.777*.、*.888*.、*.999*.
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数字的形式就是这样，最多十个数，那么，就是要在质数序列上搜索这样的数列:
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1. 用欧拉筛生成质数表（上限约100万足够）
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2. 对每个质数p：
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a. 检查是否有≥3个相同数字，且该数字∈{0,1,2}
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b. 生成所有C(n,3)种3位替换组合
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c. 对每种组合，用0-9替换，统计质数个数
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d. 若得到8个质数，返回p（首个即为答案）
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"""
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import time
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from itertools import combinations
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from math import isqrt
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from bitarray import bitarray
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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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elapsed_time = end_time - start_time
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print(f"{func.__name__} time: {elapsed_time:.6f} seconds")
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return result
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return wrapper
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def primes_list(limit: int = 10**8) -> set[int]:
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is_prime = bitarray(limit + 1)
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is_prime.setall(True)
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is_prime[:2] = False # 0和1不是素数
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# 只需筛到 sqrt(n)
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imax = isqrt(limit) + 1
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for i in range(2, imax):
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if is_prime[i]:
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# 步长i，从i*i开始（小于i*i的已被更小的素数筛过）
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is_prime[i * i : limit + 1 : i] = False
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return {i for i, isp in enumerate(is_prime) if isp}
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@timer
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def solve() -> list[int]:
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primes = primes_list(10**6)
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# 遍历所有可能的模式
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for prime in sorted(primes):
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if prime < 1000: # 从4位数开始
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continue
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s = str(prime)
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# 检查每个出现3次或以上的数字
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for d in set(s):
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positions = [i for i, c in enumerate(s) if c == d]
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if len(positions) < 3:
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continue
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# 尝试替换3个位置
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for combo in combinations(positions, 3):
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if 0 in combo and d == "0":
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continue
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family = []
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for new_d in "0123456789":
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new_num_str = "".join(
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new_d if i in combo else c for i, c in enumerate(s)
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)
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new_num = int(new_num_str)
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# 确保长度相同且是素数
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if len(str(new_num)) == len(s) and new_num in primes:
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family.append(new_num)
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if len(family) == 8:
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return sorted(family)
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return [-1]
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if __name__ == "__main__":
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result = solve()
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print(result)
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