Paper title:

Statements and open problems on decidable sets π“§βŠ†β„• that refer to the current knowledge on 𝓧

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 Landau’s conjecture states that the set π“Ÿπ’πŸ+𝟏 of primes of the form π’πŸ+𝟏 is infinite. Landau’s conjecture implies the following unproven statement 𝚽:πœπ’‚π’“π’…(π“Ÿπ’πŸ+𝟏) <πŽβ‡’π“Ÿπ’πŸ+πŸβŠ†[𝟐,(((πŸπŸ’!)!)!)!]. We heuristically justify the statement 𝚽. This justification does not yield the finiteness/infiniteness of π“Ÿπ’πŸ+𝟏. We present a new heuristic argument for the infiniteness of π“Ÿπ’πŸ+𝟏, which is not based on the statement 𝚽. The distinction between algorithms whose existence is provable in 𝒁𝑭π‘ͺ and constructively defined algorithms which are currently known inspires statements and open problems on decidable sets π“§βŠ†β„• that refer to the current knowledge on 𝓧.
Keywords: Conjecturally Infinite Sets π“§βŠ†β„•; Constructively Defined Integer 𝒏 Satisfies πœπ’‚π’“π’…(𝓧)<πŽβ‡’π“§βŠ†(βˆ’βˆž,𝒏]; Known Elements Of A Set π“§βŠ†β„• Whose Infiniteness Is False Or Unproven; Mathematical Definitions, Statements And Open Problems With Epistemic And Informal Notions; Primes Of The Form π’πŸ+𝟏; 𝓧 Is Decidable By A Constructively Defined Algorithm Which Is Currently Known.
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