The physical limits of computation inspire an open problem that concerns abstract computable sets X⊆N and cannot be formalized in mathematics as it refers to the current knowledge on X and requires that X is naturally defined

Agnieszka Kozdeba (1) , Apoloniusz Tyszka (1)

1 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, namely (1,...,1) and (f(1),...,f(9)). Let Ψ denote the statement: if a system S⊆B 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). We write down a system A⊆B of 8 equations. The statement Ψ restricted to the system A is equivalent to the statement Φ. This heuristically proves the statement Φ . This proof does not argue that card(P(n^2+1))=ω. Algorithms always terminate. Semi-algorithms may not terminate. We explain the distinction between "existing algorithms" (i.e. algorithms whose existence is provable in ZFC) and "known algorithms" (i.e. algorithms whose existence is constructive and currently known to us). Open Problem: Is there a set X⊆N that satisfies conditions (1)-(5) ? *** (1) There are many elements of X and it is conjectured that X is infinite. (2) No known proof (computer-assisted proof) shows the finiteness/infiniteness of X. No known algorithm with no inputs, whose output is unknown to us, 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 a known algorithm with no inputs that computes an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (5) There is a known and naturally defined condition C, which can be formalized in ZFC, such that for all except at most finitely many k∈N, k satisfies the condition C if and only if k∈X. The simplest known such condition C defines in N the set X. *** We define a set X⊆N. The set X satisfies conditions (1)-(5) except the requirement that X is naturally defined. The statement Φ implies that the set X={1}∪P(n^2+1) satisfies conditions (1)-(5). Proving Landau's conjecture will disprove the last two statements. No set X⊆N will satisfy conditions (1)-(4) forever, if for every algorithm with no inputs that operates on integers, at some future day, a computer will be able to execute this algorithm in 1 second or less. Physics disproves this assumption.

Mots clés: computable set X⊆N; conjecturally infinite set X⊆N; current knowledge on X; existing algorithms; known algorithms; naturally defined set X⊆N; no known algorithm with no inputs; whose output is unknown to us; decides the finiteness/infiniteness of X; no known proof (computer-assisted proof) shows the finiteness/infiniteness of X; physical limits of computation; primes of the form n^2+1

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

Agnieszka Kozdeba, Apoloniusz Tyszka. The physical limits of computation inspire an open problem that concerns abstract computable sets X⊆N and cannot be formalized in mathematics as it refers to the current knowledge on X and requires that X is naturally defined. 2020. ⟨hal-01614087v16⟩