Inspired by this observation, we consider solving the equation with Newton's method. Starting from some initial iterate , Newton's method gives the iteration
If is a rational number, then it is clear that will be rational as well, and we can write for some integers and . Plugging this in, we get the iteration
Splitting the numerator and denominator, we have the updates
Assume that is a solution to Pell's equation. Plugging in the update, we see that is also a solution:
In other words, given an initial solution to Pell's equation, e.g., , we can generate additional solutions via Newton's method applied to the root finding problem. The ratio will also provide a better and better approximation to .
Exercise 3.5 in the book derives this procedure using a more geometric approach. This derivation via Newton's method may be well-known (my number theory knowledge is very much lacking), but I thought it was an interesting connection.