Pythagorean Triplets and Pell's Equation

September 14, 2008 by zero618e

Now that I have a general theory for all 2 variable quadratic
Diophantine equations it's worth coming back to note again the weird
connection I found between certain Pythagorean Triplets and Pell's
Equation in the form
x^2 - Dy^2 = 1
when D-1 is a perfect square. For instance for D=2, I have that for
every solution of Pell's Equation you have a Pythagorean Triplet!
But the triplets are special in that with u^2 + v^2 = w^2, v = u+1.
The connection is that w is x+y from Pell's Equation.
The more general result is that u = sqrt(D-1)j, and v = j+1, while w
still equals x+y.
Intriguingly that means that proof that there are an infinite number
of solutions for certain Pell's Equations is proof that there are an
infinity of Pythagorean Triplets of a certain form!
An easy example with D=2, is x=17, y=12, where notice you are paired
with the triplet 20, 21, 29.
That is just some low-hanging fruit that I thought I'd mention. Kind
of been a whirlwind of results flowing from playing with my
Diophantine Quadratic Theorem.