feat(0047): 添加欧拉47题质因数分解与连续整数检测实现

solutions/0047.PrimesFactors/euler_47.py

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+"""
+The first two consecutive numbers to have two distinct prime factors are:
+
+14 = 2 * 7
+15 = 3 * 5
+
+The first three consecutive numbers to have three distinct prime factors are:
+
+644 = 2^2 * 7 * 23
+645 = 3 * 5 * 43
+646 = 2 * 17 * 19
+
+Find the first four consecutive integers to have four distinct prime factors each.
+What is the first of these numbers?
+"""
+
+import time
+from typing import Dict, List
+
+
+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 to run.")
+        return result
+
+    return wrapper
+
+
+def is_probable_prime(n: int) -> bool:
+    """
+    Miller-Rabin素性测试
+    对于n < 3,317,044,064,679,887,385,961,981 (3e24)，使用确定性基底集
+    """
+    if n < 2:
+        return False
+    # 小素数快速检查
+    small_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
+    for p in small_primes:
+        if n % p == 0:
+            return n == p
+
+    # 写成 d * 2^s 的形式
+    d = n - 1
+    s = 0
+    while d % 2 == 0:
+        d //= 2
+        s += 1
+
+    # 确定性基底集（对64位整数足够）
+    # 根据研究，测试 [2, 325, 9375, 28178, 450775, 9780504, 1795265022]
+    # 可以覆盖 < 2^64 的所有数
+    test_bases = [2, 325, 9375, 28178, 450775, 9780504, 1795265022]
+
+    for a in test_bases:
+        if a % n == 0:
+            continue
+        x = pow(a, d, n)
+        if x == 1 or x == n - 1:
+            continue
+        for _ in range(s - 1):
+            x = pow(x, 2, n)
+            if x == n - 1:
+                break
+        else:
+            return False
+    return True
+
+
+def factorize(n: int) -> Dict[int, int]:
+    """
+    优化版试除法因数分解
+    返回: {质因数: 指数, ...}
+
+    优化策略：
+    1. 快速排除小因子（2, 3）
+    2. 6k±1优化（跳过所有3的倍数）
+    3. 动态更新上界（随着因子被移除，sqrt(n)减小）
+    4. 对大余数进行素性预检
+    """
+    if n < 2:
+        return {}
+
+    factors = {}
+    remaining = n
+
+    # 处理负数
+    if remaining < 0:
+        factors[-1] = 1
+        remaining = -remaining
+
+    # 步骤1: 快速处理小素数（2和3）
+    for p in [2, 3]:
+        if remaining % p == 0:
+            count = 0
+            while remaining % p == 0:
+                remaining //= p
+                count += 1
+            factors[p] = count
+
+    # 步骤2: 6k±1优化
+    # 所有大于3的素数都形如 6k±1
+    # 即依次检查 5, 7, 11, 13, 17, 19...
+    # 步长模式: +2 (到6k+1), +4 (到6k+5, 即下一个6(k+1)-1)
+    divisor = 5
+    step = 2  # 交替使用 2 和 4
+
+    # 动态计算上界：只需要检查到 sqrt(remaining)
+    # 随着remaining减小，上界也减小
+    while divisor * divisor <= remaining:
+        if remaining % divisor == 0:
+            count = 0
+            while remaining % divisor == 0:
+                remaining //= divisor
+                count += 1
+            factors[divisor] = count
+            # 更新上界（优化关键！）
+
+        divisor += step
+        step = 6 - step  # 2 -> 4 -> 2 -> 4...
+
+    # 步骤3: 如果remaining > 1，说明remaining本身是质数
+    # 使用Miller-Rabin确认（对大数避免误判）
+    if remaining > 1:
+        # 对于小余数直接确认，大余数用素性测试
+        if remaining < 1_000_000 or is_probable_prime(remaining):
+            factors[remaining] = factors.get(remaining, 0) + 1
+        else:
+            # 极小概率：remaining是合数但试除法未找到因子
+            # 此时remaining必为两个大质数的乘积，且都 > sqrt(original_n)
+            # 对于这种情况，可回退到Pollard's Rho（可选）
+            factors[remaining] = 1
+
+    return factors
+
+
+def factorize_list(n: int) -> List[int]:
+    """返回展开形式的质因数列表，如 12 -> [2, 2, 3]"""
+    factors = factorize(n)
+    result = []
+    for p, exp in sorted(factors.items()):
+        result.extend([p] * exp)
+    return result
+
+
+@timer
+def main(limit: int = 4) -> None:
+    n = 1155
+    keep_ok = False
+    res = []
+    while True:
+        tl = set(factorize_list(n))
+        if len(tl) == limit:
+            res.append(n)
+            keep_ok = True
+            if len(res) == limit and keep_ok:
+                print(res)
+                break
+        else:
+            res = []
+            keep_ok = False
+        n += 1
+
+
+if __name__ == "__main__":
+    main()

solutions/0047.PrimesFactors/readme.md

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+这个问题的核心点是快速质因数分解，我使用试除法来解决的，在4质数4连续的要求下0.3秒获得解。
+在计算5质数5连续的结果时，所需时间就不那么理想了。
+
+另外我觉得获得搜索开始的起始点，可能是除了快速计算质因数外，最重要的问题了。
+我只是使用最简单的前n个素数积作为起点，似乎也不是最好的估计。
+
+这里我只是提供一个简单的假设，n质数n连续的数字可能需要从n-1和n-2这两组数的平均数位之和，作为搜索的起点.