Python/project_euler/problem_064/sol1.py

77 lines
2.0 KiB
Python
Raw Normal View History

"""
Project Euler Problem 64: https://projecteuler.net/problem=64
All square roots are periodic when written as continued fractions.
For example, let us consider sqrt(23).
It can be seen that the sequence is repeating.
For conciseness, we use the notation sqrt(23)=[4;(1,3,1,8)],
to indicate that the block (1,3,1,8) repeats indefinitely.
Exactly four continued fractions, for N<=13, have an odd period.
How many continued fractions for N<=10000 have an odd period?
References:
- https://en.wikipedia.org/wiki/Continued_fraction
"""
from math import floor, sqrt
def continuous_fraction_period(n: int) -> int:
"""
Returns the continued fraction period of a number n.
>>> continuous_fraction_period(2)
1
>>> continuous_fraction_period(5)
1
>>> continuous_fraction_period(7)
4
>>> continuous_fraction_period(11)
2
>>> continuous_fraction_period(13)
5
"""
numerator = 0.0
denominator = 1.0
root = int(sqrt(n))
integer_part = root
period = 0
while integer_part != 2 * root:
numerator = denominator * integer_part - numerator
denominator = (n - numerator**2) / denominator
integer_part = int((root + numerator) / denominator)
period += 1
return period
def solution(n: int = 10000) -> int:
"""
Returns the count of numbers <= 10000 with odd periods.
This function calls continuous_fraction_period for numbers which are
not perfect squares.
This is checked in if sr - floor(sr) != 0 statement.
If an odd period is returned by continuous_fraction_period,
count_odd_periods is increased by 1.
>>> solution(2)
1
>>> solution(5)
2
>>> solution(7)
2
>>> solution(11)
3
>>> solution(13)
4
"""
count_odd_periods = 0
for i in range(2, n + 1):
sr = sqrt(i)
if sr - floor(sr) != 0 and continuous_fraction_period(i) % 2 == 1:
count_odd_periods += 1
return count_odd_periods
if __name__ == "__main__":
print(f"{solution(int(input().strip()))}")