Solution of Pell equation over field of p-adic numbers

Question

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers using continued fraction of p-adic numbers? Anyone?

Answer 1

I’m not well-enough read to tell you about the utility of continued fractions in the $p$-adic context; all I can do is point out that you can’t expect a continued fraction of the kind you’re used to from the solution of Pell’s equation that you know.

What makes the continued fraction process work in the real case? You have a little real number $\varepsilon$ that’s between $0$ and $1$. Then you take its reciprocal, and you get a big number. Then you subtract just the right positive integer to get a new little number $\varepsilon'$. And continue.

In the $p$-adic world, you can start with your little number $\varepsilon$, and take its reciprocal to get a big number, but the whole point of the nonarchimedean metric is that no integer will be anywhere near that big number. What are you going to subtract? There’s no natural, and certainly no unique, rational number that can stand as your partial quotient, the analog of the partial quotients that you see in continued fractions.

So the moral is that if you’re going to use continued fractions for any kind of approximative expansion of a quadratic irrationality in $\mathbb Z_p$ or $\mathbb Q_p$, the expressions will look very different, and I suspect that they will not be anywhere near as useful as in the real case.