Magic Squares of Squares Modulo All
by Lee Morgenstern, January 2012.

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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