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.

# Pythagorean Triplets and Pell's Equation

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