| Paper title: |
Statements and open problems on decidable sets π§ββ 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 Landauβs conjecture states that the set πππ+π of primes of the form ππ+π is infinite. Landauβs conjecture implies the following unproven statement π½:ππππ (πππ+π) <πβπππ+πβ[π,(((ππ!)!)!)!]. We heuristically justify the statement π½. This justification does not yield the finiteness/infiniteness of πππ+π. We present a new heuristic argument for the infiniteness of πππ+π, which is not based on the statement π½. The distinction between algorithms whose existence is provable in πππͺ and constructively defined algorithms which are currently known inspires statements and open problems on decidable sets π§ββ that refer to the current knowledge on π§. |
| Keywords: | Conjecturally Infinite Sets π§ββ; Constructively Defined Integer π Satisfies ππππ (π§)<πβπ§β(ββ,π]; Known Elements Of A Set π§ββ Whose Infiniteness Is False Or Unproven; Mathematical Definitions, Statements And Open Problems With Epistemic And Informal Notions; Primes Of The Form ππ+π; π§ Is Decidable By A Constructively Defined Algorithm Which Is Currently Known. |
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