Statements and open problems on decidable sets X⊆N
Apoloniusz Tyszka
English. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω⇒P(n^2+1)⊆[2,(((24!)!)!)!]. We heuristically justify the statement Φ. This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. The distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known inspires statements on decidable sets X⊆N that refer to the current knowledge on X.
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