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  1.  42
    Two Conjectures on the Arithmetic in R and C.Apoloniusz Tyszka - 2010 - Mathematical Logic Quarterly 56 (2):175-189.
    Let G be an additive subgroup of ℂ, let Wn = {xi = 1, xi + xj = xk: i, j, k ∈ {1, …, n }}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk: i, j, k ∈ {1, …, n }}. We discuss two conjectures. If a system S ⊆ En is consistent over ℝ , then S has a real solution which consists of numbers whose absolute values belong (...)
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  2. On Sets X \Subseteq \Mathbb{N} for Which We Know an Algorithm That Computes a Threshold Number T(X) \in \Mathbb{N} Such That X is Infinite If and Only If X Contains an Element Greater Than T(X).Apoloniusz Tyszka - manuscript
    Let \Gamma_{n} denote (k-1)!, where n \in {3,...,16} and k \in {2} \cup [2^{2^{n-3}}+1,\infty) \cap N. For an integer n \in {3,...,16}, let \Sigma_n denote the following statement: if a system of equations S \subseteq {\Gamma_{n}(x_i)=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} with Gamma instead of \Gamma_n has only finitely many solutions in positive integers x_1,...,x_n, then every tuple (x_1,...,x_n) \in (N\{0})^n that solves the original system S satisfies x_1,...,x_n \leq 2^{2^{n-2}}. Our hypothesis claims that the (...)
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  3.  3
    On Subrings R of Q for Which HTP for Solutions in R has a Positive Solution If and Only If the Set of All Diophantine Equations with a Finite Number of Solutions in R is Recursively Enumerable.Agnieszka Peszek & Apoloniusz Tyszka - manuscript
    Let R be a subring of Q with or without 1, and let for every positive integer n there exists a computable surjection from N onto R^n. 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. Smory\'nski'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) Smory\'nski's theorem easily follows (...)
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