Pell's equation   (also called the Pell–Fermat equation)   is a   Diophantine equation   of the form:

   x2   -   ny2   =   1  
with integer solutions for   x   and   y,   where   n   is a given non-square positive integer.



Task requirements

   find the smallest solution in positive integers to Pell's equation for   n = {61, 109, 181, 277}.


See also

   Wikipedia entry: Pell's equation.


import math
 
def solvePell(n):
    x = int(math.sqrt(n))
    y, z, r = x, 1, x << 1
    e1, e2 = 1, 0
    f1, f2 = 0, 1
    while True:
        y = r * z - y
        z = (n - y * y) // z
        r = (x + y) // z
 
        e1, e2 = e2, e1 + e2 * r
        f1, f2 = f2, f1 + f2 * r
 
        a, b = f2 * x + e2, f2
        if a * a - n * b * b == 1:
            return a, b
 
for n in [61, 109, 181, 277]:
    x, y = solvePell(n)
    print("x^2 - %3d * y^2 = 1 for x = %27d and y = %25d" % (n, x, y))