Abstract
We say that (x, y, z) Q3 is an associative triple in a quasigroup Q(*) if (x * y) * z = x * (y * z). It is easy to show that the number of associative triples in Q is at least (Q), and it was conjectured that quasigroups with exactly (Q) associative triples do not exist when (Q) >1.
We refute this conjecture by proving the existence of quasigroups with exactly (Q) associative triples for a wide range of values (Q). Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums.