problem66_py.txt

"""
Consider quadratic Diophantine equations of the form:

x2 – Dy2 = 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:

32 – 2×22 = 1
22 – 3×12 = 1
92 – 5×42 = 1
52 – 6×22 = 1
82 – 7×32 = 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.

"""
"""
this is the the brute force solution the correct one is just an equation from wikipedia so i think this one is more interesting
from technical point of view
"""
import math
result = []
for i in range(2, 1000):
    if not math.sqrt(i).is_integer():
        for y in range(1, 10000000):
            x = math.sqrt(i * (y ** 2) + 1)
            if x == int(x):
                result.append((x, i))
                break
result = sorted(result, key = lambda s: s[0], reverse = False )
print(result)