Add solution for Project Euler problem 188 (TheAlgorithms#2880)

* Project Euler problem 188 solution

* fix superscript notation

* split out modexpt() function, and rename parameters

* Add some more doctest, and add type hints

* Add some reference links

* Update docstrings and mark helper function private

* Fix doctests and remove/improve redundant comments

* fix as per style guide

Author: darkstar

Diff for: project_euler/problem_188/__init__.py

@@ -0,0 +1,68 @@
+"""
+Project Euler Problem 188: https://projecteuler.net/problem=188
+
+The hyperexponentiation of a number
+
+The hyperexponentiation or tetration of a number a by a positive integer b,
+denoted by a↑↑b or b^a, is recursively defined by:
+
+a↑↑1 = a,
+a↑↑(k+1) = a(a↑↑k).
+
+Thus we have e.g. 3↑↑2 = 3^3 = 27, hence 3↑↑3 = 3^27 = 7625597484987 and
+3↑↑4 is roughly 103.6383346400240996*10^12.
+
+Find the last 8 digits of 1777↑↑1855.
+
+References:
+    - https://en.wikipedia.org/wiki/Tetration
+"""
+
+
+# small helper function for modular exponentiation
+def _modexpt(base: int, exponent: int, modulo_value: int) -> int:
+    """
+    Returns the modular exponentiation, that is the value
+    of `base ** exponent % modulo_value`, without calculating
+    the actual number.
+    >>> _modexpt(2, 4, 10)
+    6
+    >>> _modexpt(2, 1024, 100)
+    16
+    >>> _modexpt(13, 65535, 7)
+    6
+    """
+
+    if exponent == 1:
+        return base
+    if exponent % 2 == 0:
+        x = _modexpt(base, exponent / 2, modulo_value) % modulo_value
+        return (x * x) % modulo_value
+    else:
+        return (base * _modexpt(base, exponent - 1, modulo_value)) % modulo_value
+
+
+def solution(base: int = 1777, height: int = 1855, digits: int = 8) -> int:
+    """
+    Returns the last 8 digits of the hyperexponentiation of base by
+    height, i.e. the number base↑↑height:
+
+    >>> solution(base=3, height=2)
+    27
+    >>> solution(base=3, height=3)
+    97484987
+    >>> solution(base=123, height=456, digits=4)
+    2547
+    """
+
+    # calculate base↑↑height by right-assiciative repeated modular
+    # exponentiation
+    result = base
+    for i in range(1, height):
+        result = _modexpt(base, result, 10 ** digits)
+
+    return result
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")