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 ZFC as it refers to our current knowledge on X
Résumé
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. Open problem: Is there a set X⊆N that satisfies conditions (1)--(5)? (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 k∈N decides whether or not k∈X. (4) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n]. (5) X is simply defined and we know an algorithm such that for every input k∈N it returns the sentence "k∈X" or the sentence "k∉X" and every returned sentence is true when k is sufficiently large. The simplest known such algorithm may return a false sentence only if k is small. We prove: (i) the set X ={k∈N: (k>β) ⇒ (β,k)∩P(n^2+1) ≠ ∅} satisfies conditions (1)--(4), (ii) the set X = P(n^2+1) satisfies conditions (1)--(3) and (5,) (iii) the statement Φ implies that the set X= P(n^2+1) satisfies condition (4).
Mots clés
large number of elements of X
mathematical statement that cannot be formalized in 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
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