The Dixmier conjecture and the shape of possible counterexamples
  We establish a lower bound for the size of possible counterexamples of the Dixmier Conjecture. We prove that $B>15$, where $B$ is the minimum of the greatest common divisor of the total degrees of $P$ and $Q$, where $(P,Q)$ runs over the counterexamples of the Dixmier Conjecture. 
Jorge A. Guccione, Juan J. Guccione and Christian Valqui
math.RA