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  1. The Physical Impossibility of Machine Computations on Sufficiently Large Integers Inspires an Open Problem That Concerns Abstract Computable Sets X⊆N and Cannot Be Formalized in the Set Theory ZFC as It Refers to Our Current Knowledge on X.Apoloniusz Tyszka & Sławomir Kurpaska - manuscript
    Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Let β=(((24!)!)!)!, and let Φ denote the implication: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆(-∞,β]. We heuristically justify the statement Φ without invoking Landau's conjecture. The set X = {k∈N: (β<k) ⇒ (β,k)∩P(n^2+1) ≠ ∅} satisfies conditions (1)--(4). (1) There are a large number of elements of X and it is conjectured that X is infinite. (2) No known algorithm decides the finiteness/infiniteness of X . (3) There is (...)
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  2. 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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  3. 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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