scholarly journals The Matiyasevich Theorem. Preliminaries

2017 ◽  
Vol 25 (4) ◽  
pp. 315-322
Author(s):  
Karol Pak
Keyword(s):  
Negative Solution ◽  
Pell’S Equation ◽  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem ◽  
Diophantine Property

Summary In this article, we prove selected properties of Pell’s equation that are essential to finally prove the Diophantine property of two equations. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem.

scholarly journals Basic Diophantine Relations

2018 ◽  
Vol 26 (2) ◽  
pp. 175-181
Author(s):  
Marcin Acewicz ◽  
Karol Pąk
Keyword(s):  
Linear Equations ◽  
Second Order ◽  
Negative Solution ◽  
Hilbert's Tenth Problem ◽  
Right Hand ◽  
Hilbert’S Tenth Problem ◽  
Finite Products ◽  
The Right

Summary The main purpose of formalization is to prove that two equations ya(z)= y, y = xz are Diophantine. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem. In our previous work [6], we showed that from the diophantine standpoint these equations can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities. In this formalization, we express these relations in terms of Diophantine set introduced in [7]. We prove that these relations are Diophantine and then we prove several second-order theorems that provide the ability to combine Diophantine relation using conjunctions and alternatives as well as to substitute the right-hand side of a given Diophantine equality as an argument in a given Diophantine relation. Finally, we investigate the possibilities of our approach to prove that the two equations, being the main purpose of this formalization, are Diophantine. The formalization by means of Mizar system [3], [2] follows Z. Adamowicz, P. Zbierski [1] as well as M. Davis [4].

scholarly journals Diophantine Equations with a Finite Number of Solutions: Craig Smorynski's Theorem, Harvey Friedman's Conjecture and Minhyong Kim's Guess

2019 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Finite Number ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Negative Solution ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Hilbert's Tenth Problem ◽  
Recursively Enumerable ◽  
Hilbert’S Tenth Problem

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.

scholarly journals Diophantine sets. Preliminaries

2018 ◽  
Vol 26 (1) ◽  
pp. 81-90
Author(s):  
Karol Pąk
Keyword(s):  
Multivariate Polynomials ◽  
Negative Solution ◽  
Hilbert's Tenth Problem ◽  
Several Variables ◽  
Hilbert’S Tenth Problem

Summary In this article, we define Diophantine sets using the Mizar formalism. We focus on selected properties of multivariate polynomials, i.e., functions of several variables to show finally that the class of Diophantine sets is closed with respect to the operations of union and intersection. This article is the next in a series [1], [5] aiming to formalize the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem.

scholarly journals Diophantine Equations with a Finite Number of Solutions: Craig Smorynski's Theorem, Harvey Friedman's Conjecture and Minhyong Kim's Guess

2019 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Finite Number ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Negative Solution ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Hilbert's Tenth Problem ◽  
Recursively Enumerable ◽  
Hilbert’S Tenth Problem

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.

scholarly journals Diophantine Equations with a Finite Number of Solutions: Craig Smorynski's Theorem, Harvey Friedman's Conjecture and Minhyong Kim's Guess

2019 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Finite Number ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Negative Solution ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Hilbert's Tenth Problem ◽  
Recursively Enumerable ◽  
Hilbert’S Tenth Problem

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.

scholarly journals Diophantine Equations with a Finite Number of Solutions: Craig Smorynski's Theorem, Harvey Friedman's Conjecture and Minhyong Kim's Guess

2019 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Finite Number ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Negative Solution ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Hilbert's Tenth Problem ◽  
Recursively Enumerable ◽  
Hilbert’S Tenth Problem

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.

Hilbert's Tenth Problem

2021 ◽  
Vol 52 (2) ◽  
pp. 36-44
Author(s):  
William Gasarch
Keyword(s):  
Short Version ◽  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem ◽  
Complete History

This column is a short version of a long version of an article based on a blog. What? I give the complete history.

Hilbert's tenth problem

1975 ◽  
Vol 3 (2) ◽  
pp. 161-184 ◽  
Author(s):  
Yu. I. Manin
Keyword(s):  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem
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