Re: Prime Sums In A Grid

On 24 Feb, 22:32, bart demoen <b...@xxxxxxxxxxxxxx> wrote:
On Tue, 24 Feb 2009 12:42:05 -0800, LudovicoVan wrote:
because these improvements might allow calculating the number of
solutions for n=4 and maybe even more, so that the still open question
as to whether there is a recurrence relation for such counts might
even find an answer (with count(2)=8 and count(3)=1152

Hello Bart,

Thank you too very much for the feedback.

Ah, if you want that, be prepared for long waiting for higher inputs,
since the problem has a lot of symmetries: a solution can be transformed
in a different solution by exchanging rows and/or columns, and by
reflecting around a diogonal, very much like a solution to an N-queens
problem can, or to the latin square problem. So, your chances to find
experimentally the number of solutions for N=10 (for instance) will be
greatly increased by counting solutions that are unique up to a symmetry.

Right, I had been thinking about that, but couldn't come up with any
clever way to avoid symmetric solutions. A simple approach I could
think of was: take three distinct numbers (say, 1, 2 and 3), and
impose the contraint that they appear in a specific left-right/up-down
orientation, but that's not enough because some solutions might have
the three numbers lined up in a row, so something more clever is
needed. I'll keep thinking about this problem, but, as I was
anticipating to Markus, not until tomorrow (or maybe even later
tonight, if I am not too destroyed by the working day).

BTW, maybe it is just my program, but in the mean time I am running the
program for N=11 and N=14, and it looks as if there are no solutions.
Or do you have a proof that for any N>1 there is at least one solution ?

No, no proof so far from the guys in sci.math, and surely not from
myself. That's why I was after some more evidence. Very interesting
that you seem not to find any solution for some values of N, as my
guess was the number of solutions could only increase for increasing
values of N.