As of today, the question remains open as to whether the quaternary quartic equation
9 · (u^2 + 7 v^2)^2 − 7 · (r^2 + 7 s^2)^2 = 2 , (*)
which M. Davis put forward in 1968, has only finitely many solutions in
integers. If the answer were affirmative then—as noted by M. Davis, Yu.
V. Matiyasevich, and J. Robinson in 1976—every r.e. set would turn out
to admit a single-fold polynomial Diophantine representation.
New candidate ‘rule-them-all’ equations, constructed by the same recipe
which led to (*), are proposed in this paper.
