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Apoloniusz Tyszka

Publications and source records attributed to Apoloniusz Tyszka.

At least 19 recordsLinked to original sources
arXiv2026

A simple example of a non-recursively enumerable set $W\subseteq N$, where $W=\{\frac{1}{2}(p+q)(p+q+1)+q: (p,q\in N)\wedge Φ(p,q)\}$ and the formula $(p,q\in N)\wedge Φ(p,q)$ concerns $(N^{p+1},\{0,\ldots,q\}^{p+1})$

Let $F(x,n)$ denote the formula $$ \exists ab ~\forall i \leqslant n ~\exists swpq ~\forall jv ~\exists eg ~\{(s+w)^2+3w+s=2i ~\wedge ~\langle[j=w ~\vee ~v=q] $$ $$ \vee~[j=3i ~\wedge ~v=p+q] ~\vee ~[j=s ~\wedge ~(v=p ~\vee ~(i=n ~\wedge ~v=q+x))] $$ $$ \vee~[j=3i+1 ~\wedge ~v=pq] ~\Rightarrow ~a=v+e+ejb ~\wedge ~v+g=jb\rangle\} $$ from J. P. Jones' article in vol. 43 of J. Symbolic Logic. From the results of Jones' article, it follows that the set $J=\{n \in \mathbb{N}: \neg F(n,n)\}$ is co-recursively enumerable and not recursively enumerable. We prove that the set $$ W=\{\frac{1}{2}(p+q)(p+q+1)+q: (p,q\in \mathbb{N})~\wedge $$ $$ \forall (x_0,\ldots,x_p) \in \mathbb{N}^{p+1} ~~\exists (y_0,\ldots,y_p) \in \{0,\ldots,q\}^{p+1}$$ $$ ((\forall j,k \in \{0,\ldots,p\} ~(x_j+1=x_k \Rightarrow y_j+1=y_k))~\wedge $$ $$ (\forall i,j,k \in \{0,\ldots,p\} ~(x_i \cdot x_j=x_k \Rightarrow y_i \cdot y_j=y_k)))\} $$ is co-recursively enumerable and not recursively enumerable. Let $β:\mathbb{N}^3 \to \mathbb{N}$ denote Gödel's $β$ function. For $x_1,x_2,x_3 \in \mathbb{N}$, $β(x_1,x_2,x_3)$ equals the remainder after integer division of $x_1$ by $1+(x_3+1) \cdot x_2$. We prove that the set $W$ consists of all $n \in \mathbb{N}$ such that $$ \forall u,v \in \mathbb{N} ~\exists a,b,p,q \in \mathbb{N} ~((2n=(p+q)(p+q+1)+2q) ~\wedge ~\forall i,j,k \in \{0,\ldots,p\} $$ $$ ((β(a,b,i) \leqslant q) ~\wedge ~(β(u,v,j)+1=β(u,v,k) \Rightarrow β(a,b,j)+1=β(a,b,k)) ~\wedge $$ $$ (β(u,v,i) \cdot β(u,v,j)=β(u,v,k) \Rightarrow β(a,b,i) \cdot β(a,b,j)=β(a,b,k)))) $$ We express the above formula in Peano arithmetic.

math.NT↗
arXiv2023

A common approach to three open problems in number theory

The following system of equations {x_1 \cdot x_1=x_2, x_2 \cdot x_2=x_3, 2^{2^{x_1}}=x_3, x_4 \cdot x_5=x_2, x_6 \cdot x_7=x_2} has exactly one solution in ({\mathbb N}\{0,1})^7, namely (2,4,16,2,2,2,2). Hypothesis 1 states that if a system of equations S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,7}} \cup {2^{2^{x_j}}=x_k: j,k \in {1,...,7}} has at most five equations and at most finitely many solutions in ({\mathbb N}\{0,1})^7, then each such solution (x_1,...,x_7) satisfies x_1,...,x_7 \leq 16. Hypothesis 1 implies that there are infinitely many composite numbers of the form 2^{2^{n}}+1. Hypotheses 2 and 3 are of similar kind. Hypothesis 2 implies that if the equation x!+1=y^2 has at most 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)}. Hypothesis 3 implies that if the equation x(x+1)=y! has at most finitely many solutions in positive integers x and y, then each such solution (x,y) belongs to the set {(1,2),(2,3)}. We describe semi-algorithms sem_j (j=1,2,3) that never terminate. For every j \in {1,2,3}, if Hypothesis j is true, then sem_j endlessly prints consecutive positive integers starting from 1. For every j \in {1,2,3}, if Hypothesis j is false, then sem_j prints a finite number (including zero) of consecutive positive integers starting from 1.

math.NT↗
arXiv2019

Hilbert's 10th Problem for solutions in a subring of Q

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smoryński'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. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether or not the equation has a solution in R. We prove that a positive solution to H_{10}(R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R \subseteq Q such that there exist computable functions τ_1,τ_2:N \to Z which satisfy (\forall n \in N τ_2(n) \neq 0) \wedge ({\frac{τ_1(n)}{τ_2(n)}: n \in N}=R). This implication for R=N guarantees that Smoryński's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R=Q.

math.LO↗
arXiv2017

Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?

The height of a rational number $p/q$ is denoted by $h(p/q)$ and equals $\text{max}(|p|,|q|)$ provided p/q is written in lowest terms. The height of a rational tuple $(x_1,...,x_n)$ is denoted by $h(x_1,...,x_n)$ and equals $\text{max}(h(x_1),...,h(x_n))$. Let $G_n={x_i+1=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}$. Let $f(1)=1$, and let $f(n+1)=2^{(2^{(f(n))})}$ for every positive integer n. We conjecture: (1) if a system $S \subseteq G_n$ has only finitely many solutions in rationals $x_1,...,x_n$, then each such solution $(x_1,...,x_n)$ satisfies $h(x_1,...,x_n) \leq {1 (\text{if} n=1), 2^{(2^{(n-2)})} (\text{if} n>1)}$; (2) if a system $S \subseteq G_n$ has only finitely many solutions in non-negative rationals $x_1,...,x_n$, then each such solution $(x_1,...,x_n)$ satisfies $h(x_1,...,x_n) \leq f(2n)$. We prove: (1) both conjectures imply that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of rational solutions, if the solution set is finite; (2) both conjectures imply 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.

math.NT↗
arXiv2017

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

Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions in non-negative integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(2n). We prove: (1) the conjecture 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 conjecture implies that the question whether or not a Diophantine equation has only finitely many rational solutions is decidable with an oracle for deciding whether or not a Diophantine equation has a rational solution, (3) the conjecture implies that the question whether or not a Diophantine equation has only finitely many integer solutions is decidable with an oracle for deciding whether or not a Diophantine equation has an integer solution, (4) the conjecture implies that if a set M \subseteq N has a finite-fold Diophantine representation, then M is computable.

math.NT↗
arXiv2017

Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?

Let B_n={x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let ξ(n) denote the smallest positive integer b such that for each system S \subseteq B_n with a unique solution in positive integers x_1,...,x_n, this solution belongs to [1,b]^n. Let g(1)=1, and let g(n+1)=2^{2^{g(n)}} for every positive integer n. We conjecture that ξ(n) \leq g(2n) for every positive integer n. We prove: (1) the function ξ: N\{0}-->N\{0} is computable in the limit; (2) if a function f:N\{0}-->N\{0} has a single-fold Diophantine representation, then there exists a positive integer m such that f(n)<ξ(n) for every integer n>m; (3) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x_1,...,x_p)=0 and returns a positive integer d with the following property: for every positive integers a_1,...,a_p, if the tuple (a_1,...,a_p) solely solves the equation D(x_1,...,x_p)=0 in positive integers, then a_1,...,a_p \leq d; (4) the conjecture implies that if a set M \subseteq N has a single-fold Diophantine representation, then M is computable; (5) for every integer n>9, the inequality ξ(n)<(2^{2^{n-5}}-1)^{2^{n-5}}+1 implies that 2^{2^{n-5}}+1 is composite.

math.LO↗
arXiv2017

Is there an algorithm that decides the solvability of a Diophantine equation with a finite number of solutions?

For a positive integer n, let θ(n) denote the smallest positive integer b such that for each system S \subseteq {x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}} which has a solution in positive integers x_1,...,x_n and which has only finitely many solutions in positive integers x_1,...,x_n, there exists a solution of S in ([1,b] \cap N)^n. We conjecture that there exists an integer δ \geq 9 such that the inequality θ(n) \leq (2^{2^{n-5}}-1)^{2^{n-5}}+1 holds for every integer n \geq δ. We prove: (1) for every integer n>9, the inequality θ(n)<(2^{2^{n-5}}-1)^{2^{n-5}}+1 implies that 2^{2^{n-5}}+1 is composite, (2) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x_1,...,x_p)=0 and returns the message "Yes" or "No" which correctly determines the solvability of the equation D(x_1,...,x_p)=0 in positive integers, if the solution set is finite, (3) if a function f:N\{0} \to N\{0} has a finite-fold Diophantine representation, then there exists a positive integer m such that f(n)<θ(n) for every integer n>m.

math.NT↗
arXiv2015

On systems of Diophantine equations with a large number of integer solutions

Let E_n={x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For each integer n \geq 13, J. Browkin defined a system B_n \subseteq E_n which has exactly b_n solutions in integers x_1,...,x_n, where b_n \in N\{0} and the sequence {b_n}_{n=13}^\infty rapidly tends to infinity. For each integer n \geq 12, we define a system T_n \subseteq E_n which has exactly t_n solutions in integers x_1,...,x_n, where t_n \in N\{0} and lim_{n \to \infty} t_n/b_n=\infty.

math.NT↗
arXiv2015

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

Let f(n)=1 if n=1, 2^(2^(n-2)) if n \in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \in {6,7,8,...}. We conjecture that if a system T \subseteq {x_i+1=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in positive integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is 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. We show that if the conjecture is true, then this can be partially confirmed by the execution of a brute-force algorithm.

math.NT↗
arXiv2015

All functions g:N-->N which have a single-fold Diophantine representation are dominated by a limit-computable function f:N\{0}-->N which is implemented in MuPAD and whose computability is an open problem

Let E_n={x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For any integer n \geq 2214, we define a system T \subseteq E_n which has a unique integer solution (a_1,...,a_n). We prove that the numbers a_1,...,a_n are positive and max(a_1,...,a_n)>2^(2^n). For a positive integer n, let f(n) denote the smallest non-negative integer b such that for each system S \subseteq E_n with a unique solution in non-negative integers x_1,...,x_n, this solution belongs to [0,b]^n. We prove that if a function g:N-->N has a single-fold Diophantine representation, then f dominates g. We present a MuPAD code which takes as input a positive integer n, performs an infinite loop, returns a non-negative integer on each iteration, and returns f(n) on each sufficiently high iteration.

math.NT↗
arXiv2014

Small systems of Diophantine equations which have only very large integer solutions

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. There is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer n>=m(f), and returns a system S \subseteq E_n such that S has infinitely many integer solutions and each integer tuple (x_1,...,x_n) that solves S satisfies x_1=f(n). For each integer n>=12 we construct a system S \subseteq E_n such that S has infinitely many integer solutions and they all belong to Z^n\[-2^{2^{n-1}},2^{2^{n-1}}]^n.

math.LO↗
arXiv2014

Does there exist an algorithm which to each Diophantine equation assigns an integer which is greater than the number (heights) of integer solutions, if these solutions form a finite set?

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. If Matiyasevich's conjecture on finite-fold Diophantine representations is true, then for every computable function f:N->N there is a positive integer m(f) such that for each integer n>=m(f) there exists a system S \subseteq E_n which has at least f(n) and at most finitely many solutions in integers x_1,...,x_n. This conclusion contradicts to the author's conjecture on integer arithmetic, which implies that the heights of integer solutions to a Diophantine equation are computably bounded, if these solutions form a finite set.

math.LO↗
arXiv2014

A hypothetical upper bound for the solutions of a Diophantine equation with a finite number of solutions

We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \leq 2^{2^{n-1}}. By the conjecture, if a Diophantine equation has only finitely many solutions in integers (non-negative integers, rationals), then their heights are bounded from above by a computable function of the degree and the coefficients of the equation. The conjecture implies that the set of Diophantine equations which have infinitely many solutions in integers (non-negative integers) is recursively enumerable. The conjecture stated for an arbitrary computable bound instead of 2^{2^{n-1}} remains in contradiction to Matiyasevich's conjecture that each recursively enumerable set M \subseteq {\mathbb N}^n has a finite-fold Diophantine representation.

math.NT↗
arXiv2014

MuPAD codes which implement limit-computable functions that cannot be bounded by any computable function

For a positive integer n, let f(n) denote the smallest non-negative integer b such that for each system S \subseteq {x_k=1,x_i+x_j=x_k,x_i*x_j=x_k: i,j,k \in {1,...,n}} with a solution in non-negative integers x_1,...,x_n, there exists a solution of S in {0,...,b}^n. We prove that the function f is strictly increasing and dominates all computable functions. We present an infinite loop in MuPAD which takes as input a positive integer n and returns a non-negative integer on each iteration. Let g(n,m) denote the number returned on the m-th iteration, if n is taken as input. Then, g(n,m) \leq m-1, 0=g(n,1)<1=g(n,2) \leq g(n,3) \leq g(n,4) \leq ... and g(n,f(n)) N that cannot be bounded by any computable function. This code takes as input a non-negative integer n, immediately returns 0, and computes a system S of polynomial equations. If the loop terminates for S, then the next instruction is executed and returns ξ(n).

cs.CC↗
arXiv2014

Is there an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the number of integer solutions, if the solution set is finite?

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let f(n) denote the greatest finite total number of solutions of a subsystem of E_n in integers x_1,...,x_n. We prove: (1) the function f is strictly increasing, (2) if a non-decreasing function g from positive integers to positive integers satisfies f(n) \geq g(n) for any n, then a finite-fold Diophantine representation of g does not exist, (3) if the question of the title has a positive answer, then there is a computable strictly increasing function g from positive integers to positive integers such that f(n) \leq g(n) for any n and a finite-fold Diophantine representation of g does not exist.

math.NT↗
arXiv2012

A new characterization of computable functions

Let E_n={x_i=1, x_i+x_j=x_k, x_i*x_j=x_k: i,j,k \in {1,...,n}}. We prove: (1) there is an algorithm that for every computable function f:N-->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer n>=m(f), and returns a system S \subseteq E_n such that S is consistent over the integers and each integer tuple (x_1,...,x_n) that solves S satisfies x_1=f(n), (2) there is an algorithm that for every computable function f:N-->N returns a positive integer w(f), for which a second algorithm accepts on the input f and any integer n>=w(f), and returns a system S \subseteq E_n such that S is consistent over N and each tuple (x_1,...,x_n) of non-negative integers that solves S satisfies x_1=f(n).

math.LO↗
arXiv2012

A subset of Z^n whose non-computability leads to the existence of a Diophantine equation whose solvability is logically undecidable

For K \subseteq C, let B_n(K)={(x_1,...,x_n) \in K^n: for each y_1,...,y_n \in K the conjunction (\forall i \in {1,...,n} (x_i=1 => y_i=1)) AND (\forall i,j,k \in {1,...,n} (x_i+x_j=x_k => y_i+y_j=y_k)) AND (\forall i,j,k \in {1,...,n} (x_i*x_j=x_k => y_i*y_j=y_k)) implies that x_1=y_1}. We claim that there is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer n>=m(f), and returns a tuple (x_1,...,x_n) \in B_n(Z) with x_1=f(n). We compute an integer tuple (x_1,...,x_{20}) for which the statement (x_1,...,x_{20}) \in B_{20}(Z) is equivalent to an open Diophantine problem. We prove that if the set B_n(Z) (B_n(N), B_n(N \setminus {0})) is not computable for some n, then there exists a Diophantine equation whose solvability in integers (non-negative integers, positive integers) is logically undecidable.

math.LO↗