Can $\sqrt{k-1} + \sqrt{k+1}$ Be a Rational Number for Any Integer k?

[anemone]

Here is this week's POTW:

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Is there an integer $k$ such that $\sqrt{k-1}+\sqrt{k+1}$ is a rational number?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

No one answered last week's problem. :(

You can find the proposed solution below:

Spoiler

Suppose $\sqrt{k-1}+\sqrt{k+1}$ is rational, and thus consider:

$\begin{align*}(\sqrt{k-1}+\sqrt{k+1})(\sqrt{k+1}-\sqrt{k-1})&=-(k-1)+(k+1)\\&=2\end{align*}$

This implies $\sqrt{k+1}$ and $\sqrt{k-1}$ are rational.

We let $k+1=a^2$ and $k-1=b^2$, where $a$ and $b$ are positive integer. This gives us back $a=\dfrac{3}{2}$ and $b=\dfrac{1}{2}$. We have reached to a contradiction therefore $\sqrt{k-1}+\sqrt{k+1}$ is irrational for every integer $k\ge 1$.