Rename Project Euler directories and other dependent changes (#3300)

* Rename all Project Euler directories:

Reason:
The change was done to maintain consistency throughout the directory
and to keep all directories in sorted order.

Due to the above change, some config files had to be modified:
'problem_22` -> `problem_022`

* Update scripts to pad zeroes in PE directories

project_euler/problem_044/sol1.py

@@ -0,0 +1,47 @@
+"""
+Problem 44: https://projecteuler.net/problem=44
+
+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 solution(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.
+    >>> solution(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"{solution() = }")