Pell’s Equation, Ii

In Part I we met Pell’s equation x2−dy2 = 1 for nonsquare positive integers d. We stated Lagrange’s theorem that every Pell equation has a nontrivial solution (an integral solution besides (±1, 0)) and saw what all solutions to Pell’s equation look like if there’s a nontrivial solution. As in Part I, “solution” means integral solution. Here we will prove Lagrange’s theorem in Section 2 and show in Section 3 how to find all the solutions of a generalized Pell equation x2 − dy2 = n. Examples are in Section 4.