euler/python/e071.py

from lib_misc import gcd


def smallest_fraction_with_new_denominator(n, d, d_new):
    return (int(d_new / d * n), d_new)


def euler_071():
    """
    The idea is to find the the closest fraction to (n, d) for every potential
    new denominator d_new. We do that by extrapolating (n, d) to d_new and
    rounding n_new down. This is what smallest_fraction_with_new_denominator
    does.

    We can then check if the greatest common divisor for n_new and d_new is
    one. I though it might be necessary to search for the next smallest n_new
    if gcd(n_new, d_new) > 1, but it turned out that this is not actually
    necessary.
    """
    n = 3
    d = 7
    n_min, d_min = 0, 1
    for d_new in range(950000, 1000000):
        n_new, d_new = smallest_fraction_with_new_denominator(n, d, d_new)
        if d_new == d:
            pass
        elif gcd(n_new, d_new) == 1 and n_new / d_new > n_min / d_min:
            n_min, d_min = n_new, d_new
    return n_min


def euler_071_farey():
    """
    https://www.cut-the-knot.org/blue/Farey.shtml
    n_min must be located between 2/5 and 3/7 for F_7
    To calculate F_d where d <= 10**6 we continously calculate the
    median between 3/7 and the new fraction.
    """
    n, d = (3, 7)
    n_D, d_D = (2, 5)
    while d_D + d <= 10**6:
        n_D = n_D + n
        d_D = d_D + d
    return n_D


if __name__ == "__main__":
    print(euler_071_farey())
    print("e071.py: " + str(euler_071()))
    assert(euler_071() == 428570)