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

Sławomir Kurpaska (1) , Apoloniusz Tyszka (2)

1 Cracow Agricultural University
2 University of Agriculture in Krakow

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: (k>β) ⇒ (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 a known algorithm that for every n∈N decides whether or not n ∈ X. (4) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n]. (5) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n] and some known definition of X is much simpler than every known definition of X \ (-∞,n]. The following problem is open: Is there a set X⊆N that satisfies conditions (1)--(3) and (5)? The set X=P(n^2+1) satisfies conditions (1)--(3). The set X={k∈N : the number of digits of k belongs to P(n^2+1)} contains 10^{10^{450}} consecutive integers and satisfies conditions (1)--(3). The statement Φ implies that both sets X satisfy condition (5).

Mots clés: large number of elements of X; mathematical statement that cannot be formalized in the set theory ZFC; no known algorithm decides the finiteness/infiniteness of X; known algorithm for every n∈N decides whether or not n∈X; complexity of a mathematical definition; computable set X⊆N; current knowledge on X; explicitly known integer n bounds X from above when X is finite; infiniteness of X remains conjectured; physical impossibility of machine computations on sufficiently large integers

Domaines: Théorie des nombres [math.NT]; Logique [math.LO]

Sławomir Kurpaska, Apoloniusz Tyszka. 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. 2020. ⟨hal-01614087v9⟩