# Thread: Question to Pell Equation

1. ## Question to Pell Equation

Let $\displaystyle x,y,d \in \mathbb{N}$. For fixed $\displaystyle d > 0$ we want to find solution pairs $\displaystyle (x,y)$ to the equation $\displaystyle x^2-dy^2=1$.

Assume we have already found the solution $\displaystyle (x_1,y_1)$ so that any other solution $\displaystyle (x,y)$ has $\displaystyle y_1 < y$.

Show that any pair $\displaystyle (x_k,y_k)$ satisfying $\displaystyle x_k + y_k\sqrt{d} = (x_1 + y_1\sqrt{d})^k$ for $\displaystyle k > 1$ is another solution.

2. Originally Posted by EinStone
Let $\displaystyle x,y,d \in \mathbb{N}$. For fixed $\displaystyle d > 0$ we want to find solution pairs $\displaystyle (x,y)$ to the equation $\displaystyle x^2-dy^2=1$.

Assume we have already found the solution $\displaystyle (x_1,y_1)$ so that any other solution $\displaystyle (x,y)$ has $\displaystyle y_1 < y$.

Show that any pair $\displaystyle (x_k,y_k)$ satisfying $\displaystyle x_k + y_k\sqrt{d} = (x_1 + y_1\sqrt{d})^k$ for $\displaystyle k > 1$ is another solution.

$\displaystyle x_k^2-dy_k^2=(x_k+y_k\sqrt{d})(x_k-y_k\sqrt{d})=(x_1^2-dy_1^2)^k=1^k=1$

Tonio

3. Originally Posted by tonio
$\displaystyle x_k^2-dy_k^2=(x_k+y_k\sqrt{d})(x_k-y_k\sqrt{d})=(x_1^2-dy_1^2)^k=1^k=1$
Do you use $\displaystyle (x_k-y_k\sqrt{d}) = (x_1 - y1\sqrt{d})^k$ and if yes, why can you do it?

4. Originally Posted by EinStone
Do you use $\displaystyle (x_k-y_k\sqrt{d}) = (x_1 - y1\sqrt{d})^k$ and if yes, why can you do it?

It follows from $\displaystyle a+b\sqrt{d}=(\alpha+\beta\sqrt{d})(\gamma+\delta\s qrt{d})\Longrightarrow$ $\displaystyle a-b\sqrt{d}=(\alpha-\beta\sqrt{d})(\gamma-\delta\sqrt{d})$ and induction (you can prove this by comparing the free coefficient

and the $\displaystyle \sqrt{d}$-coefficient in both sides, remembering that $\displaystyle \{1\,,\,\sqrt{d}\}$ is a free basis of the free abelian group $\displaystyle \mathbb{Z}[\sqrt{d}]$ ).

Tonio

5. ok nice. Sorry for the break, I had some spring break . Now I want to prove:

Show that there are no other solutions except the ones proven above.

6. Originally Posted by EinStone
ok nice. Sorry for the break, I had some spring break . Now I want to prove:

Show that there are no other solutions except the ones proven above.
The proof can be found here.

7. What was Pell's full name?

8. Originally Posted by pollardrho06
What was Pell's full name?
John Pell

9. Hey thanks for the proof, but I never heard of a "convergent" and Im not too familiar with continued fractions either. It would we nice to see a proof which uses more elementary tools, so to say.

Can you help me?

10. Originally Posted by EinStone
Hey thanks for the proof, but I never heard of a "convergent" and Im not too familiar with continued fractions either. It would we nice to see a proof which uses more elementary tools, so to say.

Can you help me?
Unfortunately, my knowledge of Diophantine equations are limited. I don't know how to explain this without using continued fractions.