On the Number of Even and Odd Latin Squares of Orderp+1

Abstract It is shown that given an odd primep, the number of even latin squares of orderp+1 is not equal to the number of odd latin squares of orderp+1. This result is a special case of a conjecture of Alon and Tarsi and has implications for various other combinatorial problems, including conjectures of Rota and Dinitz. The proof counts even and odd latin squares modulop3. This counting uses properties of isotopisms, cyclic neofields, and orthomorphisms of Z p.