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.
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