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We say a positive integer $n$ is good if there exists a permutation $a_1, a_2 . . . a_n$ of $1, 2 . . . n$ such that $k+a_k$ is perfect square for all $1 \le k \le n$. Determine all the good numbers in the set $\{11, 13, 15, 17, 19\}$

I have experimented and I think I have a conjecture: $n$ is good if it is of the form $4t+1$. For example,

$$ \begin{array}{c|lcr} k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13\\ \hline a_{k} & 8 & 2 & 13 & 12 & 11 & 10 & 9 & 1 & 7 & 6 & 5 & 4 & 3\\ \end{array} $$

I think I can argue that a case such as $n=15$ doesn't work by describing how some numbers need to go into certain positions and how there is always a contradiction, but I seek a proof which covers all numbers. Can anybody think of a trick that might solve this?

I am also curious in extending this for even numbers.

EDIT As has been pointed out, $15$ is easy to achieve. In that case, my conjecture is wrong. Please do suggest other possibilities.

Plato
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1 Answers1

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For the set given $(\{11, 13, 15, 17, 19\})$, $11$ cannot be done as both $4$ and $11$ have to be sent to $5$ and all the rest can be done like so:

$13$: $$\begin{array}{c|lcr} k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13\\ \hline a_{k} & 8 & 2 & 13 & 12 & 11 & 10 & 9 & 1 & 7 & 6 & 5 & 4 & 3\\ \end{array}$$

$15$: $$\begin{array}{c|lcr} k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15\\ \hline a_{k} & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1\\ \end{array}$$

$17$: $$\begin{array}{c|lcr} k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17\\ \hline a_{k} & 3 & 7 & 6 & 5 & 4 & 10 & 2 & 17 & 16 & 15 & 14 & 13 & 12 & 11 & 1 & 9 & 8\\ \end{array}$$

$19$: $$\begin{array}{c|lcr} k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19\\ \hline a_{k} & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 16 & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 19 & 18 & 17\\ \end{array}$$

For $13$ and $15$ there may be other examples but I haven't checked. For $17$ there is only this one (which amazingly contains a $5$ cycle) and it's inverse. For $19$ there are many others but I gave this one as it has the nice property that the squares the numbers add up to increase or stay the same moving from left to right.

In general (and for much bigger numbers) I have very little idea how to show whether it's possible or not but I suspect as there are more possibilities for each number to be sent to the probability increases as the numbers get bigger.

PJF49
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