107 lines
3.3 KiB
Python
107 lines
3.3 KiB
Python
"""
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The arithmetic sequence, 1487,4817,8147, in which each of the terms increases by 3330,
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is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the
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4-digit numbers are permutations of one another.
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There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property,
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but there is one other 4-digit increasing sequence.
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What 12-digit number do you form by concatenating the three terms in this sequence?
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"""
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import time
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from logging import root
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from pathlib import Path
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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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@timer
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def solve_ultra_optimized(n: int = 4) -> list[str]:
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limit = 10**n
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mini_limit = 10 ** (n - 1) + 1
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# ========== 1. Bitset 筛法（内存高效） ==========
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is_prime = bytearray(b"\x01") * limit
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is_prime[0] = is_prime[1] = 0
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# 埃氏筛，使用切片赋值加速
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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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start = i * i
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is_prime[start:limit:i] = b"\x00" * ((limit - start - 1) // i + 1)
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# ========== 2. 按数字签名分组 ==========
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groups = {}
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for p in range(mini_limit, limit, 2): # 只遍历奇数，4位质数必为奇数
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if is_prime[p]:
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key = "".join(sorted(str(p)))
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groups.setdefault(key, []).append(p)
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# ========== 3. 组内搜索（利用模18剪枝） ==========
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res = []
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for key, group in groups.items():
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if len(group) < 3:
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continue
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group.sort()
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group_set = set(group) # 用于O(1)成员检查
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n = len(group)
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for i in range(n):
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a = group[i]
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for j in range(i + 1, n):
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b = group[j]
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d = b - a
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# 核心数学优化：公差必须是18的倍数
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# 原因：(1) 排列数字和相同 => 模9同余 => d%9==0
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# (2) 质数>2都是奇数 => 奇+偶=奇 => d%2==0
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if d % 18 != 0:
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continue
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c = b + d
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# 边界检查（利用group有序性提前终止）
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if c >= limit:
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break
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# Bitset O(1) 质数检查
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if not is_prime[c]:
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continue
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# 确认第三项在组内（数字签名匹配）
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if c in group_set:
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res.append(f"{a}-{b}-{c}:{d}")
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return res
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if __name__ == "__main__":
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try:
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n = int(input("Search digits (default is 4-digit): ") or "4")
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except ValueError:
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n = 4
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res = solve_ultra_optimized(n)
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if lr := len(res) > 10:
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head = 10
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print(
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f"Too much data, only showing the first ten rows. Other data saved in result_{n}_digit.txt."
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)
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root_path = Path(__file__).parent
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with open(f"{root_path}/result_{n}_digit.txt", "w", encoding="utf-8") as f:
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f.write("\n".join(res))
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else:
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head = lr
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for x in res[:head]:
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print(x)
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