mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-12-18 09:10:16 +00:00
24d3cf8244
* The black formatter is no longer beta
* pre-commit autoupdate
* pre-commit autoupdate
* Remove project_euler/problem_145 which is killing our CI tests
* updating DIRECTORY.md
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
69 lines
1.8 KiB
Python
69 lines
1.8 KiB
Python
"""
|
|
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 (fast exponentiation algorithm)
|
|
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() = }")
|