欧拉第44题「五边形数」完整解法

solutions/0044.PentagonNums/euler_44.py

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+"""
+Pentagonal numbers are generated by the formula, P(n) = n(3n - 1)/2.
+The first ten pentagonal numbers are:
+    1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...
+
+It can be seen that P(4) + P(7) = 22 + 70 = 92 = P(8).
+However, their difference, 70 - 22 = 48 , is not pentagonal.
+
+Find the pair of pentagonal numbers, P_j and P_k,
+for which their sum and difference are pentagonal and D = |P_j - P_k| is minimised; what is the value of D ?
+"""
+
+import time
+
+
+def timer(func):
+    def wrapper(*args, **kwargs):
+        start_time = time.time()
+        result = func(*args, **kwargs)
+        end_time = time.time()
+        print(f"{func.__name__} taken: {end_time - start_time:.6f} seconds")
+        return result
+
+    return wrapper
+
+
+def pentagonal(n):
+    return n * (3 * n - 1) // 2
+
+
+def is_pentagonal(x):
+    """检查一个数是否为五边形数"""
+    # 解方程 n(3n-1)/2 = x => 3n² - n - 2x = 0
+    # 判别式 Δ = 1 + 24x 必须是完全平方数
+    delta = 1 + 24 * x
+    sqrt_delta = int(delta**0.5)
+    if sqrt_delta * sqrt_delta != delta:
+        return False
+    # 并且 (1 + sqrt(Δ)) ≡ 0 mod 6
+    return (1 + sqrt_delta) % 6 == 0
+
+
+@timer
+def main() -> None:
+    n = 1
+    is_quit = False
+    while not is_quit:
+        D = pentagonal(n)
+        M = n * (3 * n - 1)
+
+        # 寻找M的所有因子对(u, w)
+        # 其中u = a-b, w = 3v-1, v = a+b
+        limit = int(M**0.5)
+
+        for u in range(1, limit + 1):
+            if M % u != 0:
+                continue
+
+            w = M // u
+
+            # 检查w ≡ 2 mod 3
+            if w % 3 != 2:
+                continue
+
+            # 计算v = (w + 1) / 3
+            v = (w + 1) // 3
+
+            # 检查u和v是否同奇偶
+            if (u % 2) != (v % 2):
+                continue
+
+            # 计算a和b
+            a = (v + u) // 2
+            b = (v - u) // 2
+
+            # 检查a和b是否为正整数
+            if b <= 0 or a <= b:
+                continue
+
+            # 计算P(a)和P(b)
+            Pa = pentagonal(a)
+            Pb = pentagonal(b)
+
+            # 验证Pa - Pb = P(d)
+            if Pa - Pb != D:
+                continue
+
+            # 验证Pa + Pb是否为五边形数
+            if is_pentagonal(Pa + Pb):
+                print("Answer:")
+                print(f"  a = {a}, b = {b}")
+                print(f"  P(a) = {Pa}, P(b) = {Pb}")
+                print(f"  P(a) - P(b) = {D} = P({n})")
+                print(f"  P(a) + P(b) = {Pa + Pb}")
+                print(
+                    f"  Verify that P(a) + P(b) is a pentagonal number: {is_pentagonal(Pa + Pb)}"
+                )
+                print(f"D = {D}")
+                is_quit = True
+
+        n += 1
+
+
+if __name__ == "__main__":
+    main()