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In this equation: $\sqrt{1+2x(x-1)} = K$, $K$ is an integer constant, I need to check if there exists integer $x$ satisfying this equation. Can anyone prove some lower and upper bound for $x$. Here is how I proceeded(and failed):

Obviously: $\sqrt{2x(x-1)} \approx \sqrt{1+2x(x-1)}$

$\Rightarrow \sqrt{2x(x-1)} \approx K$

$\Rightarrow \sqrt{x(x-1)} \approx K/\sqrt{2}$

Also obvious is: $x - 1 \approx x$

$\Rightarrow x(x-1) \approx x \times x$

$\Rightarrow \sqrt{x(x-1)} \approx x$

Combining $\sqrt{x(x-1)} \approx K / \sqrt{2}$ and $\sqrt{x(x-1)} \approx x$ implies $x \approx K / \sqrt{2}$. But that doesn't give any specifig upper/lower bound.

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I'm not sure if it's promising to try to find a numerical bound rather than, as you have, a bound in terms of $K$. Also, sorry if you already know this, but you can find the solution for $x$ in terms of $K$ using algebra. – 000 May 18 '12 at 22:27
We are solving $2x^2-2x+1=K^2$ or equivalently $4x^2-4x+2=2K^2$ or equivalently $(2x-1)^2=2K^2-1$. Not many candidates! – André Nicolas May 18 '12 at 22:34
So basically $2K^2 - 1$ has to be a perfect square. Some possibilities are $K=1$, $K=5$, $K=17$. Any other solutions? – Minh May 18 '12 at 22:40
@Minh Yes $K = 29, K = 169, K = 985$ etc. BTW, $K = 17$ is not a valid solution. – f.nasim May 18 '12 at 22:46
@Everyone In fact I need to find such $K$ for which $x > 10^{12}$. I thought I could avoid squaring by finding a bound for $x$. But now seems anyway I need to use high-precision arithmetic. – f.nasim May 18 '12 at 22:59
up vote 7 down vote accepted

We are solving $2x^2-2x+1=K^2$ or equivalently $4x^2-4x+2=2K^2$ or equivalently $(2x-1)^2=2K^2-1$. For any explicit $K$, there are not many candidates!

If we are interested in the general theory, we might note that we are dealing here with the (negative) Pell Equation $$u^2-2v^2=-1.$$ There is a complete characterization of all the solutions. Let $n$ be an odd integer, and let $u_n+v_n\sqrt{2}=(1+\sqrt{2})^n$. Then $(u_n,v_n)$ is a solution, and we obtain all positive solutions in this way.

One can also get a nice explicit recurrence, with integer coefficients, for the $u_n$ and for the $v_n$. This can be derived from the fact that $$u_{n+2}+v_{n+2}\sqrt{2}=(3+2\sqrt{2})(u_n+v_n\sqrt{2}).$$

Alternately, let $(s_k,t_k)$ be the $k$-th positive solution of $s^2-2t^2=-1$. A bit of playing shows that each of $(s_k)$ and $(t_k)$ satisfies the recurrence $$w_{k+2}=6w_{k+1}-w_k.$$ The only difference is in the initial conditions.

Remark: One can find bounds, indeed exact formulas, for the $k$-th positive solution of the equation $s^2-2t^2=-1$. For if we call this $(s_k,t_k)$, then $s_k+t_k\sqrt{2}=(1+\sqrt{2})^{2k-1}$. It follows that $s_k-t_k\sqrt{2}=(1-\sqrt{2})^{2k-1}$. By adding, we find that $$2s_k=(1+\sqrt{2})^{2k-1}+(1-\sqrt{2})^{2k-1}.$$ One can do slightly better, by noting that $(1-\sqrt{2})^{2k-1}$ is negative, and small in absolute value. Thus $$2s_k=\lfloor(1+\sqrt{2})^{2k-1}\rfloor.$$ One can in a similar way obtain an explicit formula for $t_k$.

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+1 I was reading about the Pell Equation today. Nice generalization. – 000 May 18 '12 at 22:41
@André Nicolas I'm not familiar with Pell equations, need to study them. I got how you build the recurrence from $u_n + v_n\sqrt{2} = (1 + \sqrt{2})^n$ and indeed it works. I verified some of them. It think $(v_n, u_n)$ is a solution rather than $(u_n, v_n)$. Still I get no idea for answer to my original question. – f.nasim May 19 '12 at 19:43
Sorry I was wrong. $(u_n, v_n)$ is right solution. – f.nasim May 19 '12 at 19:56
@sven.hedin: Just compute using one of the recurrences I have given. Our numbers grow by a factor of about $5.8$ each time. With that kind of growth, it doesn't take long to get into the range you want. Or else you can use the formula involving the exponential and greatest integer function that I added. The downside is that we then need high precision arithmetic, while for the recurrence approach it is only integer arithmetic. – André Nicolas May 19 '12 at 20:22

Solving the quadratic gives $x = \frac{1 \pm \sqrt{2k^2-1}}{2}$. The bounds are clear from this.

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