“One equation to rule them all”, revisited

If the quaternary quartic equation 9 · (u² + 7v²)² − 7 · (r² + 7s²)² = 2 (*) which M. Davis put forward in 1968 has only finitely many solutions in integers, then — it was observed by M. Davis, Yu. V. Matiyasevich, and J. Robinson in 1976 — every listable set would turn out to admit a single-fold Diophantine representation. In 1995, D. Shanks and S. S. Wagstaff conjectured that (*) has infinitely many solutions; while in doubt, it seemed wise to us to single out new candidates for the role of “rule-them-all equation”. We offer three new quaternary quartic equations, each obtained by much the same recipe which led to (*). The significance of those can be supported by arguments analogous to the ones found in Davis’s original paper; moreover, they might play a key role in settling the conjecture that every listable set has a single-fold (or, at least, a finite-fold) representation. Directly from the unproven assertion that any of the novel equations has only finitely many solutions in integers, one can construct a Diophantine relation of exponential growth, as we show in detail for one, namely 3 · (r² + 3s²)² − (u² + 3v²)² = 2, of the new candidate rule-them-all equations. An account of Julia Robinson’s earliest Diophantine reduction of exponentiation to any relation of exponential growth is also included, for the sake of self-containedness.