euler0066.py

#!/usr/bin/python
r"""Diophantine equation

    Consider quadratic Diophantine equations of the form:

    $$x^2 - Dy^2 = 1$$

    For example, when $D=13$, the minimal solution in $x$ is $649^2 - 13\times
    180^2 = 1$.

    It can be assumed that there are no solutions in positive integers when $D$
    is square.

    By finding minimal solutions in $x$ for $D = \{2, 3, 5, 6, 7\}$, we obtain
    the following:

    $$3^2 - 2\times 2^2 = 1$$
    $$2^2 - 3\times 1^2 = 1$$
    $$[9]^2 - 5\times 4^2 = 1$$
    $$5^2 - 6\times 2^2 = 1$$
    $$8^2 - 7\times 3^2 = 1$$

    Hence, by considering minimal solutions in $x$ for $D\leq 7$, the largest
    $x$ is obtained when $D=5$.

    Find the value of $D\leq 1000$ in minimal solutions of $x$ for which the
    largest value of $x$ is obtained."""

from __future__ import division
from math       import sqrt, floor
from eulerlib   import convergents

__tags__ = ["Pell's equation", 'maximization', 'quadratic equations']

def sqrtCF(d):
    """Returns the simple continued fraction representation of ``sqrt(d)``"""
    sqrtD = sqrt(d)
    P = 0
    Q = 1
    while True:
        a = int(floor((P + sqrtD) / Q))
        yield a
        P = a * Q - P
        Q = (d - P*P) // Q  # It can be shown that Q evenly divides d - P*P

squares = set(i*i for i in range(32))  # all perfect squares <= 1000

def solve():
    maxX = 0
    maxD = 0
    for d in xrange(2, 1001):
        if d in squares:
            continue
        for (x,y) in convergents(sqrtCF(d)):
            if x*x - d * y * y == 1:
                if x > maxX:
                    maxX = x
                    maxD = d
                break
    return maxD

if __name__ == '__main__':
    print solve()