Create reverse_factorial_recursive.py

maths/reverse_factorial_recursive.py

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+def reverse_factorial_recursive(value: int, current_divisor: int = 1) -> int:
+    """
+    Return x such that x! == value, otherwise return -1.
+
+    The function divides `value` by 1, 2, 3, ... recursively. If the repeated
+    division reduces `value` exactly to 1, the factorial root x is
+    (current_divisor - 1). If the division ever has a remainder, no integer x
+    exists and the function returns -1.
+
+    Parameters
+    ----------
+    value : int
+        The positive integer to test (candidate factorial value).
+    current_divisor : int, optional
+        The current divisor used while reducing `value` (default is 1).
+
+    Returns
+    -------
+    int
+        The factorial root (x) if x! == value, otherwise -1.
+
+    Examples
+    --------
+    >>> reverse_factorial_recursive(120)
+    5
+    >>> reverse_factorial_recursive(24)
+    4
+    >>> reverse_factorial_recursive(150)
+    -1
+    >>> reverse_factorial_recursive(1)
+    1
+    >>> reverse_factorial_recursive(2)
+    2
+    """
+    if not isinstance(value, int):
+        raise TypeError("value must be an integer")
+    if not isinstance(current_divisor, int):
+        raise TypeError("current_divisor must be an integer")
+
+    if value < 1:
+        raise ValueError("value must be a positive integer")
+
+    # Special-case: initial call with value == 1 should return 1 (since 1! = 1).
+    if value == 1 and current_divisor == 1:
+        return 1
+
+    # If value reduced to 1 during recursion, the factorial root is divisor - 1.
+    if value == 1:
+        return current_divisor - 1
+
+    # If not divisible by the current divisor, it's not a factorial number.
+    if value % current_divisor != 0:
+        return -1
+
+    return reverse_factorial_recursive(value // current_divisor, current_divisor + 1)