Conjecturally computable functions which unconditionally do not have any finite-fold Diophantine representation

Tyszka, Apoloniusz

doi:10.1016/j.ipl.2013.07.004

Cited by 10 publications

(5 citation statements)

References 3 publications

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“…, a n+11 ). Hence, Theorem 1 disproves the conjecture in [10], where the author proposed the upper bound 2 2 n−1 for positive integer solutions to any system…”

Section: Theoremsupporting

confidence: 58%

All Functions $$g: \mathbb{N} \rightarrow \mathbb{N}$$ Which have a Single-Fold Diophantine Representation are Dominated by a Limit-Computable Function $$f: \mathbb{N}\setminus \{0\} \rightarrow \mathbb{N}$$ Which is Implemented in MuPAD and Whose Computability is an Open Problem

Tyszka

2015

Computation, Cryptography, and Network Security

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“…, a n+11 ). Hence, Theorem 1 disproves the conjecture in [10], where the author proposed the upper bound 2 2 n−1 for positive integer solutions to any system…”

Section: Theoremsupporting

confidence: 58%

All Functions $$g: \mathbb{N} \rightarrow \mathbb{N}$$ Which have a Single-Fold Diophantine Representation are Dominated by a Limit-Computable Function $$f: \mathbb{N}\setminus \{0\} \rightarrow \mathbb{N}$$ Which is Implemented in MuPAD and Whose Computability is an Open Problem

Tyszka

2015

Computation, Cryptography, and Network Security

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“…Proof. By Lemma 26, we take k = 2 as the initial value of k. The following MuPAD code {1, 2, 3, 4, 5, 6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,42,43,44,45,46,47,48,49 Definition. For a positive integer n, by a program of length n we understand any sequence of terms x 1 , .…”

Section: Letmentioning

confidence: 99%

On Sets ${\mathcal X} \subseteq$ $\mathbb N$ for Which We Know an Algorithm That Computes a Threshold Number $t({\mathcal X}) \in$ $\mathbb N$ Such That ${\mathcal X}$ Is Infinite If and Only If ${\mathcal X}$ Contains an eLement Greater Than $t({\mathcal X})$

Tyszka¹

2019

Preprint

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Let $\Gamma_{\fbox{n}}(k)$ denote $(k-1)!$, where $n \in \{3,\ldots,14\}$ and $k \in \{2\} \cup \left\{2^{\textstyle 2^{n-3}}+1, 2^{\textstyle 2^{n-3}}+2, 2^{\textstyle 2^{n-3}}+3, \ldots\right\}$. For an integer $n \in \{3,\ldots,14\}$, let $\Sigma_n$ denote the following statement: if a system of equations ${\mathcal S} \subseteq \{\Gamma_{\fbox{n}}(x_i)=x_k:~i,k \in \{1,\ldots,n\}\} \cup \{x_i \cdot x_j=x_k:~i,j,k \in \{1,\ldots,n\}\}$ has only finitely many solutions in positive integers $x_1,\ldots,x_n$, then each such solution $(x_1,\ldots,x_n)$ satisfies $x_1,\ldots,x_n \leqslant 2^{\textstyle 2^{n-2}}$. The statement $\Sigma_6$ proves the following implication: if the equation $x(x+1)=y!$ has only finitely many solutions in positive integers $x$ and $y$, then each such solution $(x,y)$ belongs to the set $\{(1,2),(2,3)\}$. The statement $\Sigma_6$ proves the following implication: if the equation $x!+1=y^2$ has only finitely many solutions in positive integers $x$ and $y$, then each such solution $(x,y)$ belongs to the set $\{(4,5),(5,11),(7,71)\}$. The statement $\Sigma_9$ implies the infinitude of primes of the form $n^2+1$. The statement $\Sigma_9$ implies that any prime of the form $n!+1$ with $n \geqslant 2^{\textstyle 2^{9-3}}$ proves the infinitude of primes of the form $n!+1$. The statement $\Sigma_{14}$ implies the infinitude of twin primes.

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“…Proof. By Lemma 19, we take k = 2 as the initial value of k. The following MuPAD code {1, 2, 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36, 37, 38,…”

Section: Letmentioning

confidence: 99%

On Sets ${\mathcal X} \subseteq$ $\mathbb N$ for Which We Know an Algorithm That Computes a Threshold Number $t({\mathcal X}) \in$ $\mathbb N$ Such That ${\mathcal X}$ Is Infinite If and Only If ${\mathcal X}$ Contains an eLement Greater Than $t({\mathcal X})$

Tyszka¹

2018

Preprint

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We define computable functions $g,h:$ $\mathbb N$ $\setminus \{0\} \to$ $\mathbb N$ $\setminus \{0\}$. For an integer $n \geqslant 3$, let $\Psi_n$ denote the following statement: if a system ${\mathcal S} \subseteq \Bigl\{x_i!=x_k: (i,k \in \{1,\ldots,n\}) \wedge (i \neq k)\Bigr\} \cup \Bigl\{x_i \cdot x_j=x_k: i,j,k \in \{1,\ldots,n\}\Bigr\}$ has only finitely many solutions in positive integers $x_1,\ldots,x_n$, then each such solution $(x_1,\ldots,x_n)$ satisfies $x_1,\ldots,x_n \leqslant g(n)$. For a positive integer $n$, let $\Gamma_n$ denote the following statement: if a system $S \subseteq \Bigl\{x_i \cdot x_j=x_k:~i,j,k \in \{1,\ldots,n\}\Bigr\} \cup \Bigl\{2^{\textstyle 2^{\textstyle x_i}}=x_k:~i,k \in \{1,\ldots,n\}\Bigr\}$ has only finitely many solutions in positive integers $x_1,\ldots,x_n$, then each such solution $(x_1,\ldots,x_n)$ satisfies $x_1,\ldots,x_n \leqslant h(n)$. We prove: (1) if the equation $x!+1=y^2$ has only finitely many solutions in positive integers, then the statement $\Psi_6$ guarantees that each such solution $(x,y)$ belongs to the set $\{(4,5),(5,11),(7,71)\}$, (2) the statement $\Psi_9$ proves the following implication: if there exists a positive integer $x$ such that $x^2+1$ is prime and $x^2+1>g(7)$, then there are infinitely many primes of the form $n^2+1$,  (3) the statement $\Psi_9$ proves the following implication: if there exists an integer $x \geqslant g(6)$ such that $x!+1$ is prime, then there are infinitely many primes of the form $n!+1$, (4) the statement $\Psi_{16}$ proves the following implication: if there exists a twin prime greater than $g(14)$, then there are infinitely many twin primes, {\bf (5)}~the statement $\Gamma_{13}$ proves the following implication: if $n \in$ $\mathbb N$ $\setminus \{0\}$ and $2^{\textstyle 2^{\textstyle n}}+1$ is composite and greater than $h(12)$, then $2^{\textstyle 2^{\textstyle n}}+1$ is composite for infinitely many positive integers $n$.

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Conjecturally computable functions which unconditionally do not have any finite-fold Diophantine representation

Cited by 10 publications

References 3 publications

All Functions $$g: \mathbb{N} \rightarrow \mathbb{N}$$ Which have a Single-Fold Diophantine Representation are Dominated by a Limit-Computable Function $$f: \mathbb{N}\setminus \{0\} \rightarrow \mathbb{N}$$ Which is Implemented in MuPAD and Whose Computability is an Open Problem

All Functions $$g: \mathbb{N} \rightarrow \mathbb{N}$$ Which have a Single-Fold Diophantine Representation are Dominated by a Limit-Computable Function $$f: \mathbb{N}\setminus \{0\} \rightarrow \mathbb{N}$$ Which is Implemented in MuPAD and Whose Computability is an Open Problem

On Sets ${\mathcal X} \subseteq$ $\mathbb N$ for Which We Know an Algorithm That Computes a Threshold Number $t({\mathcal X}) \in$ $\mathbb N$ Such That ${\mathcal X}$ Is Infinite If and Only If ${\mathcal X}$ Contains an eLement Greater Than $t({\mathcal X})$

On Sets ${\mathcal X} \subseteq$ $\mathbb N$ for Which We Know an Algorithm That Computes a Threshold Number $t({\mathcal X}) \in$ $\mathbb N$ Such That ${\mathcal X}$ Is Infinite If and Only If ${\mathcal X}$ Contains an eLement Greater Than $t({\mathcal X})$

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