scholarly journals A hypothetical way to compute an upper bound for the heights of solutions of a Diophantine equation with a finite number of solutions

2015 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Number Of Solutions

scholarly journals A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

2018 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Rational Solution ◽  
Infinitely Many Solutions ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions

Let f ( 1 ) = 1 , and let f ( n + 1 ) = 2 2 f ( n ) for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn ≤ f (2n). We prove:   (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, then M is computable.

scholarly journals A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

2018 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Rational Solution ◽  
Infinitely Many Solutions ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions

Let f ( 1 ) = 1 , and let f ( n + 1 ) = 2 2 f ( n ) for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn ≤ f (2n). We prove:   (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, then M is computable.

scholarly journals A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

2018 ◽  
Vol 8 (1) ◽  
pp. 109-114
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Rational Solution ◽  
Computable Function ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions ◽  
Positive Integers

Abstract We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, . . . , xi, then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

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.

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.

SOME QUARTIC DIOPHANTINE EQUATIONS IN THE GAUSSIAN INTEGERS

2015 ◽  
Vol 92 (2) ◽  
pp. 187-194
Author(s):  
FARZALI IZADI ◽  
RASOOL FOROOSHANI NAGHDALI ◽  
PETER GEOFF BROWN
Keyword(s):  
Finite Number ◽  
Elliptic Curve ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Number Of Solutions ◽  
Gaussian Integers

In this paper we examine solutions in the Gaussian integers to the Diophantine equation $ax^{4}+by^{4}=cz^{2}$ for different choices of $a,b$ and $c$. Elliptic curve methods are used to show that these equations have a finite number of solutions or have no solution.

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