euler/e026.py

# -*- coding: utf-8 -*-
"""Find the value of d < 1000 for which 1/d contains the longest recurring cycle.A unit fraction contains 1 in the numerator.

The decimal representation of the unit fractions with denominators 2 to 10 are given:
    1/2	= 	0.5
    1/3	= 	0.(3)
    1/4	= 	0.25
    1/5	= 	0.2
    1/6	= 	0.1(6)
    1/7	= 	0.(142857)
    1/8	= 	0.125
    1/9	= 	0.(1)
    1/10	= 	0.1
Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.

Find the value of d < 1000 for which 1/d contains the longest recurring cycle in its decimal fraction part.
"""

from decimal import Decimal
from e003 import pfactor

def coprime(a, b):
    """Are integers a and b coprime?"""

    primes_a = pfactor(a)
    primes_b = pfactor(b)
    primes = [p for p in primes_a if p in primes_b]
    return bool(primes)

def long_division(dividend, divisor):
    dividend = str(dividend).replace('.', '')
    i = -1
    remainder = 0
    while True:
        while remainder < divisor:
            i += 1
            remainder *= 10
            try:
                remainder += int(dividend[i])
            except:
                pass
            if remainder == 0:
                return
        digit = str(remainder // divisor)
        remainder = remainder % divisor
        yield digit, remainder

def get_cycle_length(numerator, denominator):
    """Determine the period length of a non-terminating decimal"""

    length = None
    numerator = Decimal(numerator)
    denominator = Decimal(denominator)

    if denominator == 1:
        return 1
    
    """
    From Wikipedia:
        Terminating decimals represent rational numbers of the form
        k/2^n5^m

        However, a terminating decimal also has a representation as a
        repeating decimal, obtained by decreasing the final (nonzero)
        digit by one and appending an infinitely repeating sequence of
        nines. 1 = 0.999999... and 1.585 = 1.584999999... are two
        examples of this.
    """
    if coprime(denominator, 10):
        return 1

    digits = []
    for digit in long_division(numerator, denominator):
        if digit in digits:
            cycle = digits[digits.index(digit):]
            return len(cycle)
        digits.append(digit)

    else:
        raise Exception('Non-reciprocal period detection not yet implemented')
    return length

def main():
    MAX = 1000
    longest_cycle = (0, 0)
    for i in xrange((MAX), 1, -1):
        # The period of 1/k for integer k is always <= k - 1
        if longest_cycle[1] >= i:
            break
        
        cycle = get_cycle_length(1, i)
        if longest_cycle[1] < cycle:
            longest_cycle = (i, cycle)
    (i, cycle) = longest_cycle
    print 'Longest cycle length is', cycle, 'for i =', i
if __name__ == '__main__':
    main()