Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

# Diophantine Equations with a Finite Number of Solutions: Craig Smorynski's Theorem, Harvey Friedman's Conjecture and Minhyong Kim's Guess

Version 1 : Received: 16 February 2019 / Approved: 18 February 2019 / Online: 18 February 2019 (10:34:28 CET)
Version 2 : Received: 18 February 2019 / Approved: 19 February 2019 / Online: 19 February 2019 (11:16:51 CET)
Version 3 : Received: 25 February 2019 / Approved: 25 February 2019 / Online: 25 February 2019 (15:09:05 CET)
Version 4 : Received: 1 March 2019 / Approved: 1 March 2019 / Online: 1 March 2019 (12:58:53 CET)

How to cite: Tyszka, A. Diophantine Equations with a Finite Number of Solutions: Craig Smorynski's Theorem, Harvey Friedman's Conjecture and Minhyong Kim's Guess. Preprints 2019, 2019020156 (doi: 10.20944/preprints201902.0156.v3). Tyszka, A. Diophantine Equations with a Finite Number of Solutions: Craig Smorynski's Theorem, Harvey Friedman's Conjecture and Minhyong Kim's Guess. Preprints 2019, 2019020156 (doi: 10.20944/preprints201902.0156.v3).

## Abstract

Matiyasevich's theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non-negative integers. Smorynski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smorynski's theorem easily follows from Matiyasevich's theorem, (2 ) Hilbert's Tenth Problem for Q has a negative solution if and only if the set of all Diophantine equations with a finite number of rational solutions is not recursively enumerable.

## Keywords

Davis-Putnam-Robinson-Matiyasevich theorem, Diophantine equation which has at most finitely many solutions in \mbox{non-negative} integers, Hilbert's Tenth Problem, Hilbert's Tenth Problem for Q, Matiyasevich's theorem, recursively enumerable set, Smorynski's theorem

## Subject

MATHEMATICS & COMPUTER SCIENCE, Logic

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