✨ 新增 0039、0040 题解及配套文档

solutions/0039.IntegerTriangles/euler_39.py

@@ -6,3 +6,98 @@ If p is the perimeter of a right angle triangle with integral length sides,
 
 For which value of p<=1000, is the number of solutions maximised?
 """
+
+import time
+import math
+
+
+def timer(func):
+    def wrapper(*args, **kwargs):
+        start_time = time.time()
+        result = func(*args, **kwargs)
+        end_time = time.time()
+        print(f"{func.__name__} took time: {end_time - start_time:.6f} seconds")
+        return result
+    return wrapper
+
+
+def generate_pythagorean_triples(max_perimeter: int = 1000) :
+    """
+    使用yield生成勾股三元组的m和n参数
+
+    参数:
+        max_perimeter: 三角形周长的最大值
+
+    生成:
+        (m, n) 元组，满足：
+        - m > n > 0
+        - m和n互质 (gcd(m, n) == 1)
+        - m和n一奇一偶
+        - 三角形周长 2*m*(m+n) < max_perimeter
+    """
+    # 从m=2开始（因为m>n≥1）
+    m = 2
+
+    while True:
+        # 对于每个m，n从1到m-1
+        for n in range(1, m):
+            # 检查是否满足互质条件
+            if math.gcd(m, n) != 1:
+                continue
+
+            # 检查是否一奇一偶
+            if (m % 2) == (n % 2):
+                continue
+
+            # 计算对应三角形的周长
+            perimeter = 2 * m * (m + n)
+
+            # 如果周长超过限制，则停止生成
+            if perimeter >= max_perimeter:
+                # 对于当前m，如果最小周长（当n=1时）已经超过限制，
+                # 那么后续更大的m也会超过，所以完全停止
+                if n == 1:
+                    return
+
+            # 如果周长在限制内，则yield当前(m, n)
+            if perimeter < max_perimeter:
+                yield (m, n)
+
+        m += 1
+
+
+def triangle_length(m:int, n:int) -> int:
+    return 2 * m * (m + n)
+
+def get_abc(m:int, n:int) -> tuple[int, int, int]:
+    a = m**2 - n**2
+    b = 2 * m * n
+    c = m**2 + n**2
+    return a, b, c
+
+def primitive_triangles(limit: int = 1000) -> dict[int, list] :
+    triangle_lengths = {}
+    for m, n in generate_pythagorean_triples(limit):
+        length = triangle_length(m, n)
+        if length not in triangle_lengths:
+            triangle_lengths[length] = set()
+        a,b,c = get_abc(m, n)
+        triangle_lengths[length].add((a, b, c))
+        for k in range(2, limit // length + 1) :
+            lenp = k * (a + b + c)
+            if lenp not in triangle_lengths:
+                triangle_lengths[lenp] = set()
+            triangle_lengths[lenp].add((a*k, b*k, c*k))
+    return triangle_lengths
+
+
+@timer
+def main():
+    max_perimeter = 1000
+    triangle_lengths = primitive_triangles(max_perimeter)
+    print(triangle_lengths)
+    max_length = max(triangle_lengths, key=lambda x: len(triangle_lengths[x]))
+    print(f"Max length: {max_length}, Count: {len(triangle_lengths[max_length])}")
+
+if __name__ == "__main__":
+    main()