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It is my understanding that it has not yet been determined if it is possible to construct a $3$x$3$ magic square where all the entries are squares of integers. Is this correct? Has any published work been done on this problem?

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The existence or not of a non-trivial integer 3x3 magic square of squares is STILL a unsolved problem.

The quoted reference to Kevin Brown's web pages only discusses an extremely special configuration of numbers, which does not exist. The page does NOT claim to prove non-existence for all possible magic squares.

If you are interested in this topic you should consult the web-site

http://www.multimagie.com/

which gives lots of details and references.

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There is a claimed proof of nonexistence by J. C. Ferreira, but is it correct? https://arxiv.org/abs/1506.06621

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    $\begingroup$ Crackpot alert! Clicked on his name and I can't take this guy seriously when he publishes 20 versions of his Riemann Hypothesis "proof" arxiv.org/abs/math/0405531 $\endgroup$
    – qwr
    Apr 18, 2018 at 4:41
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    $\begingroup$ @qwr The error is equation (41). When Ferreira substitutes $c=(n+w)^2$ into equation (37), they replace $n$ with $(n+w)^2$ with $(n+w)^2$. Obviously if you let $n=(n+w)^2$ it follows that $w=0$. This is probably a recreational mathematician who spent so many hours staring at the problem that they managed to delude themselves into thinking they had a proof. $\endgroup$
    – user377597
    Nov 9, 2018 at 23:41

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