Paper title:

# Statements and open problems on decidable sets π§ββ that refer to the current knowledge on π§

DOI: https://doi.org/10.4316/JACSM.202202005
Published in: Issue 2, (Vol. 16) / 2022
Publishing date: 2022-10-11
Pages: 31-35
Author(s): TYSZKA Apoloniusz
Abstract. Edmund Landauβs conjecture states that the set πππ+π of primes of the form ππ+π is infinite. Landauβs conjecture implies the following unproven statement π½:ππππ(πππ+π) <πβπππ+πβ[π,(((ππ!)!)!)!]. We heuristically justify the statement π½. This justification does not yield the finiteness/infiniteness of πππ+π. We present a new heuristic argument for the infiniteness of πππ+π, which is not based on the statement π½. The distinction between algorithms whose existence is provable in ππ­πͺ and constructively defined algorithms which are currently known inspires statements and open problems on decidable sets π§ββ that refer to the current knowledge on π§.
Keywords: Conjecturally Infinite Sets π§ββ; Constructively Defined Integer π Satisfies ππππ(π§)<πβπ§β(ββ,π]; Known Elements Of A Set π§ββ Whose Infiniteness Is False Or Unproven; Mathematical Definitions, Statements And Open Problems With Epistemic And Informal Notions; Primes Of The Form ππ+π; π§ Is Decidable By A Constructively Defined Algorithm Which Is Currently Known.
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