Add solution for Project Euler problem 135 (#4035)

project_euler/problem_135/sol1.py

@@ -0,0 +1,61 @@
+"""
+Project Euler Problem 135: https://projecteuler.net/problem=135
+
+Given the positive integers, x, y, and z,
+are consecutive terms of an arithmetic progression,
+the least value of the positive integer, n,
+for which the equation,
+x2 − y2 − z2 = n, has exactly two solutions is n = 27:
+
+342 − 272 − 202 = 122 − 92 − 62 = 27
+
+It turns out that n = 1155 is the least value
+which has exactly ten solutions.
+
+How many values of n less than one million
+have exactly ten distinct solutions?
+
+
+Taking x,y,z of the form a+d,a,a-d respectively,
+the given equation reduces to a*(4d-a)=n.
+Calculating no of solutions for every n till 1 million by fixing a
+,and n must be multiple of a.
+Total no of steps=n*(1/1+1/2+1/3+1/4..+1/n)
+,so roughly O(nlogn) time complexity.
+
+"""
+
+
+def solution(limit: int = 1000000) -> int:
+    """
+    returns the values of n less than or equal to the limit
+    have exactly ten distinct solutions.
+    >>> solution(100)
+    0
+    >>> solution(10000)
+    45
+    >>> solution(50050)
+    292
+    """
+    limit = limit + 1
+    frequency = [0] * limit
+    for first_term in range(1, limit):
+        for n in range(first_term, limit, first_term):
+            common_difference = first_term + n / first_term
+            if common_difference % 4:  # d must be divisble by 4
+                continue
+            else:
+                common_difference /= 4
+                if (
+                    first_term > common_difference
+                    and first_term < 4 * common_difference
+                ):  # since x,y,z are positive integers
+                    frequency[n] += 1  # so z>0 and a>d ,also 4d<a
+
+    count = sum(1 for x in frequency[1:limit] if x == 10)
+
+    return count
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")