The <a href="https://gilkalai.wordpress.com/2017/03/15/test-your-intuition-about-the-alon-tarsi-conjecture/">Alon&ndash;Tarsi conjecture</a> says that if $n$ is even, then the number of even Latin squares is different from the number of odd Latin squares (where the parity of a Latin square can be defined as the product of the parities of its rows and columns, considered as permutations).  It is known in special cases, in particular when $n = p\pm 1$, but is open in general. The proof techniques for $n=p\pm 1$ make use of special properties of primes. For example, Drisko's proof of the case $n=p+1$ uses the Sylow theorems; in the Sylow theorems, you cannot simply replace the prime $p$ with an arbitrary integer.