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# A New Proof of Smoryński’s Theorem

Version 1
: Received: 26 September 2017 / Approved: 26 September 2017 / Online: 26 September 2017 (04:44:11 CEST)

Version 2 : Received: 27 September 2017 / Approved: 27 September 2017 / Online: 27 September 2017 (04:05:45 CEST)

Version 2 : Received: 27 September 2017 / Approved: 27 September 2017 / Online: 27 September 2017 (04:05:45 CEST)

How to cite:
Tyszka, A. A New Proof of Smoryński’s Theorem. *Preprints* **2017**, 2017090127 (doi: 10.20944/preprints201709.0127.v2).
Tyszka, A. A New Proof of Smoryński’s Theorem. Preprints 2017, 2017090127 (doi: 10.20944/preprints201709.0127.v2).

## Abstract

We prove: (1) the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable, (2) the set of all Diophantine equations which have at most finitely many solutions in positive integers is not recursively enumerable, (3) the set of all Diophantine equations which have at most finitely many integer solutions is not recursively enumerable, (4) analogous theorems hold for Diophantine equations

*D*(*x*_{1}, …,*x*) = 0, where_{p}*p*∈ N\{0} and for every*i*∈ {1, …,*p*} the polynomial*D*(*x*_{1}, …,*x*) involves a monomial_{p}*M*with a non-zero coefficient such that*x*divides_{i}*M*, (5) the set of all Diophantine equations which have at most*k*variables (where*k*≥ 9) and at most finitely many solutions in non-negative integers is not recursively enumerable.## Keywords

Davis-Putnan Robinson-Matiyasevich theorem; Diophantine equation which has at most finitely many solutions in non-negative integers; Diophantine equation which has at most finitely many solutions in positive integers; Diophantine equation which has at most finitely many integer solutions; Hilbert’s Tenth Problem; Matiyasevich’s theorem; recursively enumerable set; Smoryński’s theorem

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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