The physical limits of computation inspire an open problem that concerns decidable sets X⊆N and cannot be formalized in ZFC as it refers to the current knowledge on X

Agnieszka Kozdęba (1, 2) , Apoloniusz Tyszka (2)

1 Faculty of Mathematics and Computer Science of the Jagiellonian University
2 Université d'Agriculture Hugo Kollataj

Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. 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)⊆(-∞,f(7)]. Let B denote the system of equations: {x_i!=x_k: i,k∈{1,...,9}}∪{x_i⋅x_j=x_k: i,j,k∈{1,...,9}}. We write down a system U⊆B of 9 equations which has exactly two solutions in positive integers x_1,...,x_9, namely (1,...,1) and (f(1),...,f(9)). We write down a system A⊆B of 8 equations. Let Λ denote the statement: if the system A has at most finitely many solutions in positive integers x_1,...,x_9, then each such solution (x_1,...,x_9) satisfies x_1,...,x_9≤f(9). The statement Λ is equivalent to the statement Φ. This heuristically proves the statement Φ . This proof does not yield that card(P(n^2+1))=ω. The following problem is open: Is there a set X⊆N such that (there is a constructively defined integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]) ∧ (X is decidable by a constructively defined algorithm) ∧ (there are many elements of X) ∧ (the infiniteness of X is conjectured and cannot be decided by any known method) ∧ (X has the simplest definition among known sets Y⊆N with the same set of known elements)? Let F(X) denote the conjunction of the first four conditions of the problem. The set X={k∈N: (k>f(7)) ⇒ (f(7),k)∩P(n^2+1)≠∅} satisfies the formula F(X). No set X⊆N will satisfy the formula F(X) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove this assumption. The set X=P(n^2+1) satisfies the conjunction of the last four conditions of the problem. The statement Φ implies that X=P(n^2+1) solves the problem. It seems that the conjunction from the problem implies that the set X is naturally defined, where this term has only informal meaning.

Mots clés: constructively defined algorithm; constructively defined integer n satisfies card(X)<ω ⇒ X⊆(-∞; n]; current knowledge on a set X⊆N; decidable and conjecturally infinite set X⊆N; distinction between existing algorithms and known algorithms; known elements of a set X⊆N; physical limits of computation; primes of the form n^2+1

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

Agnieszka Kozdęba, Apoloniusz Tyszka. The physical limits of computation inspire an open problem that concerns decidable sets X⊆N and cannot be formalized in ZFC as it refers to the current knowledge on X. 2021. ⟨hal-01614087v21⟩