I'm reading (the rather shrot paper) The nonexistence of certain finite projective planes by Bruck and Ryser, in which they prove that
$\quad$ If $N$ is $1$ or $2$ modulo $4$, and if the square free part of $N$ contains at least one prime factor of the form $4k + 3$, then there does not exist a finite projective plane geometry with $N + 1$ points on a line.
They introduce some matrix invariants (under congruence) and reach the following identity $$c_p(B)=(-1,N)_p^{\frac{N(N+1)}{2}}=+1$$ for every odd prime $p$ and some matrix $B$. This identity is not satisfied by the $N$ in the theorem statement.
Here $c_p$ is the Hasse-Minkowski invariant, defined through the Hilbert norm-residue symbols $(a,b)_p$
My question is: could this method be improved to show something stronger? It seems to me that some ingenious play with $c_p$ and the other invariants, one could get many more conditions similar to the one in the theorem statement.
This is the first time I see a real interplay between the size of a matrix and some of its properties. Are there other theorems which rely on this specific relationships? By this I mean "if a matrix $A$ has such and such properties, then it cannot have size $n$" and not the usual "the determinant/trace depends on the size"
Thanks in advance!