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  1. Two Conjectures on the Arithmetic in ℝ and ℂ†.Apoloniusz Tyszka - 2010 - Mathematical Logic Quarterly 56 (2):175-184.
    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 to (...)
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  2. On Sets X \Subseteq N Whose Conjectural Finiteness (Infiniteness) Inspires Semi-Formally Stated Open Problems Related to Fundamental Mathematical Concepts.Apoloniusz Tyszka - manuscript
    Let P_{n^2+1} denote the set of all primes of the form n^2+1, and let M denote the set of all positive multiples of elements of the set P_{n^2+1} \cap ((((24!)!)!)!,\infty). We prove that the set X={0,...,(((24!)!)!)!} \cup M satisfies the following conditions: (1) card(X) is greater than a big positive integer and it is conjectured that X is infinite, (2) we do not know any algorithm deciding the finiteness of X, (3) a known and short algorithm for every n \in (...)
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  3. Hilbert's 10th Problem for Solutions in a Subring of Q.Agnieszka Peszek & Apoloniusz Tyszka - 2019 - Scientific Annals of Computer Science 29 (1):101-111.
    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 (...)
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