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In one of my previous questions, I sought the solution to the most elementary magic square. This time, I seek an answer to a much more complicated case. I seek solutions for the following magic square(represented as a matrix): $$\begin{bmatrix}a^n & b^n & c^n\\\ d^n & e^n & f^n\\\ g^n & h^n & i^n\end{bmatrix}$$ Such that:

  1. Sum of all elements is identical in all rows, columns and diagonals.
  2. $a,b,...,i$ are all distinct positive integers.
  3. $n>2$

Being completely honest, this problem has stumped me. Since there is no restriction on integers(in my previous question, only 1-9 were the permissible integers) and further, exponentiation is involved, I cannot even start using any brute-force method. I'd like to get some hints so as to at least get started with a solution. Further, I'm sceptical of $a,b,...,i$ being all positive integers- I feel this task might be achievable if negative integers are allowed(in that case, odd powers could lead to reduction of sums with other positive entries). In case it can be conclusively proven that the above magic square has no solutions for positive integers, does it become solvable over all integers?

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    $\begingroup$ I'm pretty sure even the $n=2$ case is still open. $\endgroup$
    – Josh B.
    Aug 8, 2020 at 18:02
  • $\begingroup$ @JoshB. I only realised that after doing some web search upon posting this question. Nevertheless, can it be conclusively proven that for distinct positive integers, this magic square has no solutions? $\endgroup$
    – Manan
    Aug 9, 2020 at 4:25

1 Answer 1

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Some hints:

  1. If $S$ is the common sum of the rows, columns and diagonals, show that $e^n=S/3$.
  2. Infer that $a^n+i^n=2e^n$.
  3. Then refer to this paper by Darmon & Merel, and in particular to Main Theorem 1 on page 2.

Notes

  1. Although steps 1 and 2 are elementary, the proof of the Darmon & Merel result is definitely not.
  2. There is some confusion about the date of the Darmon & Merel paper, see explanation here.
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  • $\begingroup$ Thank you! I'll look into the paper- I hope it doesn't need any background beyond elementary number theory? $\endgroup$
    – Manan
    Aug 10, 2020 at 17:18
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    $\begingroup$ @Manan It's not difficult to understand the statement of the theorem. Understanding the proof is another matter. $\endgroup$ Aug 10, 2020 at 18:34

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