If you're pointing out that my argument isn't rigorous, I know. It can be overcome by some kind of non-random conspiracy among primes. But it needs to be a hell of a strong conspiracy, much stronger than what you mention. Even if the whole square had to consist of only 1 3 7 9, you'd still have 4^(N^2) possible squares, and 1/N^(2N) of them would still be a huge number.
Example, just for fun:
9 7 9 7 7 9 9 1
1 7 9 9 7 1 3 1
7 9 3 9 7 3 9 9
3 3 1 7 9 1 9 7
7 3 3 1 9 7 3 1
7 7 9 1 7 1 3 9
3 1 1 7 9 1 1 9
3 9 3 3 3 3 1 1
Heck, I can even make these:
1 1 1 9 1 9 9 1
9 1 1 1 1 1 9 9
1 9 1 9 9 1 1 9
1 1 1 1 1 9 1 1
1 9 1 1 9 9 1 1
9 1 9 1 1 1 1 9
9 9 1 1 9 1 9 1
1 9 1 9 9 1 1 9
Bottom line, primes are much more common than you think :-)
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