Nested radical expressions involving the terms $1,2,3,..,n$

Question

Find all natural numbers $n$ for which there is a permutation $\sigma$ of set {$1,2,...,n$} such that:$$\sqrt{\sigma(1) + \sqrt{\sigma(2) + \sqrt{... + \sqrt{\sigma(n)}}}}$$ is a rational number. I am pretty sure that the only solutions are $n = 1, 3$. I have also noticed that if: $$a_i = \sqrt{\sigma(i) + \sqrt{... + \sqrt{\sigma(n)}}} \text{ and if } a_1 ∈ Q \Rightarrow a_2,..,a_n ∈ Q\Rightarrow a_1,..,a_n ∈ Z.$$

Answer 1

1. Observe that if a square root is irrational, adding natural numbers and taking more square roots would never return it to being rational, hence each of the intermediate results (your $a_i$, that is) must be an integer. Upd. OK, you've done that already while I was typing.
2. Establish the maximum value ($M$) for the intermediate results. An upper bound of $\sqrt{n+\sqrt{n+\sqrt{n+\dots}}}$ (which, BTW, may be turned into a closed-form expression) would suffice, I think.
3. Find the greatest square below $n$ (say, $k^2$).
4. Consider the integer $(k-1)^2+1$. It is someplace there in the permutation, so it should be added to something that would bring it to the next nearest square (at least).
5. Show that (for sufficiently large $n$) your $M$ is too small for that to happen.