Created problem_44 in project_euler (#2348)

* Create __int__.py

* Update and rename project_euler/__int__.py to project_euler/problem_44/__int__.py

* Add files via upload

* Update sol1.py

* Update __int__.py

* Delete __int__.py

* Create __init__.py

* Update project_euler/problem_44/sol1.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update project_euler/problem_44/sol1.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update project_euler/problem_44/sol1.py

Co-authored-by: Christian Clauss <cclauss@me.com>

Co-authored-by: Christian Clauss <cclauss@me.com>

project_euler/problem_44/__init__.py

@@ -0,0 +1 @@
+#

project_euler/problem_44/sol1.py

@@ -0,0 +1,45 @@
+"""
+Pentagonal numbers are generated by the formula, Pn=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 P4 + P7 = 22 + 70 = 92 = P8. However, their difference,
+70 − 22 = 48, is not pentagonal.
+
+Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference
+are pentagonal and D = |Pk − Pj| is minimised; what is the value of D?
+"""
+
+
+def is_pentagonal(n: int) -> bool:
+    """
+    Returns True if n is pentagonal, False otherwise.
+    >>> is_pentagonal(330)
+    True
+    >>> is_pentagonal(7683)
+    False
+    >>> is_pentagonal(2380)
+    True
+    """
+    root = (1 + 24 * n) ** 0.5
+    return ((1 + root) / 6) % 1 == 0
+
+
+def compute_num(limit: int = 5000) -> int:
+    """
+    Returns the minimum difference of two pentagonal numbers P1 and P2 such that
+    P1 + P2 is pentagonal and P2 - P1 is pentagonal.
+    >>> compute_num(5000)
+    5482660
+    """
+    pentagonal_nums = [(i * (3 * i - 1)) // 2 for i in range(1, limit)]
+    for i, pentagonal_i in enumerate(pentagonal_nums):
+        for j in range(i, len(pentagonal_nums)):
+            pentagonal_j = pentagonal_nums[j]
+            a = pentagonal_i + pentagonal_j
+            b = pentagonal_j - pentagonal_i
+            if is_pentagonal(a) and is_pentagonal(b):
+                return b
+
+
+if __name__ == "__main__":
+    print(f"{compute_num() = }")