Not so easy! You have to prove, amongst other things that there's a prime twixt n^2 and n^2 + n. We don't know for sure that there's one twixt n^2 and (n+1)^2. R. On Wed, 4 Jun 2003, James Propp wrote:
I should mention that the conjecture is true for all n between 2 and 100.
Moreover, if one looks at the minimum (over k) of the number of primes in the kth row of the n-by-n square, one gets the sequence
1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 3, 4, 4, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 4, 4, 5, 4, 4, 5, 5, 2, 6, 6, 5, 4, 6, 4, 5, 7, 7, 3, 7, 8, 4, 5, 10, 7, 5, 6, 5, 5, 10, 7, 8, 8, 6, 10, 7, 5, 5, 8, 7, 7, 5, 10, 7, 8, 10, 7, 7, 10, 10, 9, 12, 7, 11, 10, 10, 9, 7, 13, 11, 10, 10, 11, 10, 11, 10, 11, 12, 11, 8, 11, 9
which shows steady increase (up to slight fluctuations). So I suspect that the conjecture would yield to a two-pronged attack (brute force calculations for small n, rigorous asymptotics for large n).
Jim Propp
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