# Diophantine equation (Euler Problem 66)¶

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Consider quadratic Diophantine equations of the form:

$x^2 – Dy^2 = 1$

For example, when D=13, the minimal solution in x is 6492 – 13×1802 = 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:

$3^2 – 2×2^2 = 1$

$2^2 – 3×1^2 = 1$

$9^2 – 5×4^2 = 1$

$5^2 – 6×2^2 = 1$

$8^2 – 7×3^2 = 1$

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

Find the value of D ≤ 1000 in minimal solutions of x for which the largest value of x is obtained.

In [1]:

    import math

    def get_floor_sqrt(n):
        return math.floor(math.sqrt(n))

    assert(get_floor_sqrt(5) == 2)
    assert(get_floor_sqrt(23) == 4)

    def next_expansion_1(current_a, current_nominator, current_denominator, original_number):
        cn = current_nominator
        cd = current_denominator
        cn, cd = cd, (original_number - cd * cd) // cn
        next_a = math.floor((math.sqrt(original_number) + cn) // cd)
        cn = cn - next_a * cd
        cn *= -1

        return next_a, cn, cd

    def get_continued_fraction_sequence(n):

        # If number is a square number we return it.
        floor_sqrt = get_floor_sqrt(n)
        if n == floor_sqrt * floor_sqrt:
            return ((floor_sqrt), [])

        # Otherwise, we calculate the next expansion till we
        # encounter a step a second time.
        a = floor_sqrt
        cn = a
        cd = 1
        sequence = []
        previous_steps = []

        while not (a, cn, cd) in previous_steps :
            #print("a: {} cn: {} cd: {}".format(a, cn, cd))
            previous_steps.append((a, cn, cd))
            a, cn, cd = next_expansion_1(a, cd, cn, n)
            sequence.append(a)
        sequence.pop()
        return ((floor_sqrt), sequence)

    def get_continued_fraction_sequence(n):

        # If number is a square number we return it.
        floor_sqrt = get_floor_sqrt(n)
        if n == floor_sqrt * floor_sqrt:
            return ((floor_sqrt), [])

        # Otherwise, we calculate the next expansion till we
        # encounter a step a second time.
        a = floor_sqrt
        cn = a
        cd = 1
        sequence = []
        previous_steps = []

        while not (a, cn, cd) in previous_steps :
            #print("a: {} cn: {} cd: {}".format(a, cn, cd))
            previous_steps.append((a, cn, cd))
            a, cn, cd = next_expansion_1(a, cd, cn, n)
            sequence.append(a)
        sequence.pop()
        return ((floor_sqrt), sequence)

    assert(get_continued_fraction_sequence(1) == ((1), []))
    assert(get_continued_fraction_sequence(4) == ((2), []))
    assert(get_continued_fraction_sequence(25) == ((5), []))
    assert(get_continued_fraction_sequence(2) == ((1), [2]))
    assert(get_continued_fraction_sequence(3) == ((1), [1,2]))
    assert(get_continued_fraction_sequence(5) == ((2), [4]))
    assert(get_continued_fraction_sequence(13) == ((3), [1,1,1,1,6]))

In [43]:

    def gcd(a, b):
        if b > a:
            a, b = b, a
        while a % b != 0:
            a, b = b, a % b
        return b

    def add_fractions(n1, d1, n2, d2):
        d = d1 * d2
        n1 = n1 * (d // d1)
        n2 = n2 * (d // d2)
        n = n1 + n2
        p = gcd(n, d)
        return (n // p, d // p)

    def next_expansion_2(previous_numerator, previous_denumerator, value):
        if previous_numerator == 0:
            return (value, 1)
        return add_fractions(previous_denumerator, previous_numerator, value, 1)

    def get_fractions(n, x):
        #print("get_fractions(n={}, x={})".format(n, x))
        # Get sequence of a_x
        first_value, sequence = get_continued_fraction_sequence(n)
        sequence = [first_value] + math.ceil(x / len(sequence)) * sequence
        sequence = sequence[:x + 1]
        sequence = sequence[::-1]

        n, d = 0, 1
        for s in sequence:
            n, d = next_expansion_2(n, d, s)
        return (n, d)

    assert(get_fractions(7, 0) == (2, 1))
    assert(get_fractions(7, 1) == (3, 1))
    assert(get_fractions(7, 2) == (5, 2))
    assert(get_fractions(7, 3) == (8, 3))

In [48]:

    def get_minimal_solution(d):
        for i in range(0, 100):
            x, y = get_fractions(d, i)
            if x * x - d * y * y == 1:
                return((x, y))

    assert(get_minimal_solution(2) == (3, 2))
    assert(get_minimal_solution(3) == (2, 1))
    assert(get_minimal_solution(85) == (285769, 30996))
    assert(get_minimal_solution(109) == (158070671986249, 15140424455100))

In [52]:

    from math import sqrt

    def is_square(n):
        return sqrt(n).is_integer()

    print(is_square(4))

    True

In [55]:

    x_max = 0
    d_max = 0

    for d in range(2, 1001):
        if is_square(d):
            continue

        x, y = get_minimal_solution(d)
        if x > x_max:
            x_max = x
            d_max = d

    print("d: {} x: {}".format(d_max, x_max))
    s = d_max
    assert(s == 661)

    d: 661 x: 16421658242965910275055840472270471049

In [ ]: