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
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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