Pell's Equation - Transformations

Transformations

I. The related equation,

(eq.2)

can be used to find solutions to the positive Pell equation for certain d. Legendre proved that all primes of form d = 4m + 3 solve one case of eq.2, with the form 8m + 3 solving the negative, and 8m + 7 for the positive. Their fundamental solution then leads to the one for x2−dy2 = 1. This can be shown by squaring both sides of eq. 2,

to get,

Since from eq.2, then,

or simply,

showing that fundamental solutions to eq.2 are smaller than eq.1. For example, u2-3v2 = -2 is {u,v} = {1,1}, so x2 − 3y2 = 1 has {x,y} = {2,1}. On the other hand, u2 − 7v2 = 2 is {u,v} = {3,1}, so x2 − 7y2 = 1 has {x,y} = {8,3}.

II. Another related equation,

(eq.3)

can also be used to find solutions to Pell equations for certain d, this time for the positive and negative case. For the following transformations, if fundamental {u,v} are both odd, then it leads to fundamental {x,y}.

1. If u2 − dv2 = −4, and {x,y} = {(u2 + 3)u/2, (u2 + 1)v/2}, then x2 − dy2 = −1.

Ex. Let d = 13, then {u,v} = {3, 1} and {x,y} = {18, 5}.

2. If u2 − dv2 = 4, and {x,y} = {(u2 − 3)u/2, (u2 − 1)v/2}, then x2 − dy2 = 1.

Ex. Let d = 13, then {u,v} = {11, 3} and {x,y} = {649, 180}.

3. If u2 − dv2 = −4, and {x,y} = {(u4 + 4u2 + 1)(u2 + 2)/2, (u2 + 3)(u2 + 1)uv/2}, then x2 − dy2 = 1.

Ex. Let d = 61, then {u,v} = {39, 5} and {x,y} = {1766319049, 226153980}.

Especially for the last transformation, it can be seen how solutions to {u,v} are much smaller than {x,y}, since the latter are sextic and quintic polynomials in terms of u.