Always Show Your Work

Dear JC

I know how to solved the Pell's equations

x^2 - by^2 = 1

and

x^2 - by^2 = - 1

, but I am interested in the more general

x^2 - by^2 = c

Now I know if we have one solution

(p,q)

then we can find infinitely many solutions with

(p - q\sqrt {b})(r - s\sqrt {b})^n

where

(r,s)

is the first solution to

x^2 - by^2 = 1

My question is, how do we know when we have all the "primitive" solutions

(p,q)

that will generate all the solutions? (Also, is it possible that there would be infinite primitive solutions?)

Sincerely,

Badly Learning Odd Wacked Math Equations




Dear Badly,
NERD!!!!

Ever Patient and Kind,
Jesus H. Christ