51题完成解决

solutions/0051.PrimeDigitReplace/euler_51_better.py

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