How find all positive real $\beta$ such A finite number of $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Question

Define sequence $\{a_{n}\}$,such $$a_{1}=1,a_{2}=2,a_{k+2}=2a_{k+1}+a_{k},k\ge 1$$

Find all positive real number $\beta$,such only have a finite number of relatively prime integers $(p,q)$ such that $$\left|\dfrac{p}{q}-\sqrt{2}\right|<\dfrac{\beta}{q^2}$$ and there can't exsit $n$ such $q=a_{n}$

and this problem is Germany National Olympiad 2013 last problem (2), see: http://www.mathematik-olympiaden.de/aufgaben/52/4/A52124b.pdf

My try: since $$ a_{n+2}=2a_{n+1}+a_{n}\Longrightarrow r^2=2r+1\Longrightarrow r_{1}=\sqrt{2}+1,r_{2}=1-\sqrt{2}$$ so $$a_{n}=A(1+\sqrt{2})^{n-1}+B(1-\sqrt{2})^{n-1},a_{1}=1,a_{2}=2$$ so $$A+B=1,A(1+\sqrt{2})+B(1-\sqrt{2})=2\Longrightarrow A=\dfrac{2+\sqrt{2}}{4},B=\dfrac{2-\sqrt{2}}{4}$$ so $$a_{n}=\dfrac{2+\sqrt{2}}{4}(1+\sqrt{2})^n+\dfrac{2-\sqrt{2}}{4}(1-\sqrt{2})^n$$ so there can't postive integer $n$,such $q=a_{n}$

other hand $$\left|\frac{p}{q}-\sqrt{2}\right| = \frac{|p^2-2q^2|}{q^2\left|\frac{p}{q} + \sqrt{2}\right|}$$ Now we can't use the pell equation,because this problem is finite number of relatively prime integers $(p,q)$

so how solve it?

I have read this solution,It's Nice: How find the value $\beta$ such $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

I think we can find other methods to solve this problem?

This is different problem

I don't know why,and it is said china all can't comment.maybe this post explain why?so I have say in there http://meta.math.stackexchange.com/questions/16661/mse-requires-javascript-from-another-domain-which-is-blocked-or-failed-to-load

Answer 1

According to your computations, $$ \left|\frac{p}{q}-\sqrt2\right| = \frac{\left|\frac{p^2}{q^2}-\sqrt2^2\right|}{\frac{p}{q}+\sqrt2} = \frac1{q^2}\cdot\frac{|p^2-2q^2|}{2\sqrt2+\Big(\frac{p}{q}-\sqrt2\Big)}. $$

The condition $q\ne a_n$ excludes the solutions of the Pellian equation $p^2-2q^2=\pm1$. But allows $p^2-2q^2=2$.

I. First we show that there are only finitely many good pairs $(p,q)$ for $\beta<\frac1{\sqrt2}$.

Suppose that $\beta<\frac1{\sqrt2}$, and let $\varepsilon=\frac2\beta-2\sqrt2$. Consider a good pair $(p,q)$. If $q>\sqrt{\beta/\varepsilon}$ then $|\frac{p}{q}-\sqrt2|\le \frac{\beta}{q^2}<\varepsilon$, and thus $$ \frac\beta{q^2} \ge \left|\frac{p}{q}-\sqrt2\right| \ge \frac1{q^2}\cdot\frac{|p^2-2q^2|}{2\sqrt2+\Big|\frac{p}{q}-\sqrt2\Big|} > \frac1{q^2}\cdot\frac2{2\sqrt2+\varepsilon} = \frac\beta{q^2}, $$ contradiction. Hence, the set of possible values $q$ is bounded by $\sqrt{\beta/\varepsilon}$ and thus finite.

II. Now we construct infinitely many good pairs ($p,q)$ for $\beta\ge\frac1{\sqrt2}$.

The Pellian equation $p^2-2q^2=2$ has infinitely many solutions; for such pairs $\frac{p}{q}>\sqrt2$, so $$ \left|\frac{p}{q}-\sqrt2\right| = \frac1{q^2}\cdot\frac{|p^2-2q^2|}{2\sqrt2+\Big(\frac{p}{q}-\sqrt2\Big)} < \frac1{q^2}\cdot\frac2{2\sqrt2+0} \le \frac\beta{q^2}. $$

Therefore, the answer is $\beta\in\left(0,\frac1{\sqrt2}\right)$.

Answer 2

Not a complete answer, but I will show that if $(p,q)$ are solution to $|p^2 - 2q^2| = 1$ then $q = a_n$ for some $n$ (as Jyrki said in the comments).

The fundamental solution of the Pell equation $|p^2 - 2q^2| = 1$ is $(p,q) = (1,1)$. All the other solutions can therefore be written on the form

$$p_m + q_m \sqrt{2} = (\sqrt{2} + 1)^m$$

From this we can extract a recursion formula for $(p_m,q_m)$. We have

$$p_{m+1} + q_{m+1} \sqrt{2} = (\sqrt{2} + 1)(p_{m}+ q_{m} \sqrt{2})$$

Multiplying out and using that since $p_m,q_m$ are integers and $\sqrt{2}$ irrational ($A_n + \sqrt{2}B_n = 0 \implies A_n=B_n=0$ if $A_n,B_n$ are integers) then

$$p_{m+1} - p_m = 2q_m$$ $$q_{m+1} - q_{m} = p_{m}$$

Using $(q_{m+2} - q_{m+1}) - (q_{m+1} - q_{m}) = p_{m+1}-p_m = 2q_n$ we can eliminate $p_m$ to get a recursion realation for $q_m$ only

$$q_{m+2} = 2q_{m+1} + q_m$$

which is the same equation as for $a_n$. From $q_1 = p_1 = 1$ we get $q_2 = 2$ so the initial conditions are also the same.