Brute force

The problem is to find the value of \( D \leq 1000 \) for which \( x \) is maximised in the equation \( x^2 - Dy^2 = 1 \).

Brute forcing the solution is easy, for each \( D \leq 1000 \) that is not a perfect square, and for each \( x > 1 \), if \( y = \sqrt{\frac{x^2 - 1}{D}} \) is an integer, then \( x^2 - Dy^2 = 1 \) is satisfied.

From solution1.py:

def diophantine_equation():
    res = 0
    for d in range(2, 1001):
        if sqrt(d).is_integer():
            continue

        for x in itertools.count(2):
            if (sqrt((x**2 - 1) / d)).is_integer():
                res = max(res, x)
                break

    return res

However, the brute force approach is not effective and does not provide the solution for \( D = 61 \) as \( x \) becomes too large...