How not to prove the Alon-Tarsi conjecture

The sign of a Latin square is -1 if it has an odd number of rows and columns that are odd permutations; otherwise, it is +1. Let L-n(E) and L-n(O) be, respectively, the number of Latin squares of order n with sign +1 and -1. The Alon-Tarsi conjecture asserts that L-n(E)not equal L-n(O) when n is even. Drisko showed that L-p+1(E) not equivalent to L-p+1(O) (mod p(3)) for prime p >= 3 and asked if similar congruences hold for orders of the form p(k) + 1, p + 3, or pq + 1. In this article we show that if t