**Magic Squares of Squares Modulo All**

Choose any set of odd primes.

Multiply them all together
to get M.

Then a 3x3 magic square of distinct entries, where each entry is
a quadratic residue of every chosen prime, can be specified as follows.

8M+1 M+1 6M+1

3M+1 5M+1
7M+1

4M+1 9M+1 2M+1

Every entry produces 1 as a remainder when divided by any
chosen prime.

1 is a quadratic residue of any odd prime.

There are 9 terms in arithmetic progression, so no consecutive
4 of them can all be squares.

So this can never produce a solution.

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