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“One equation to rule them all”, revisited
Cantone, Domenico
Omodeo, Eugenio G.
2021
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e-ISSN
2464-8728
Abstract
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.
Publisher
EUT Edizioni Università di Trieste
Source
Domenico Cantone, Eugenio G. Omodeo, "'One equation to rule them all', revisited" in: "Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.53 (2021)", EUT Edizioni Università di Trieste, Trieste, 2021. pp.
Languages
en
Rights
Attribution-NonCommercial-NoDerivatives 4.0 Internazionale
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