Is there a computable and widely known set X⊆N whose infiniteness remains conjectured and unproven, although a constructively defined integer n satisfies ω>card(X) ⇒ X⊆(-∞,n]?
Résumé
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)). 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 yield that card(P(n^2+1))=ω. Algorithms always 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). Conditions (1)-(5) concern sets X⊆N. *** (1) There are many elements of X and it is conjectured that X is infinite. (2) No known algorithm with no input returns the logical value of the statement card(X)=ω. (3) A known algorithm for every k∈N decides whether or not k∈X. (4) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (5) There is a known formula φ(x) of Peano arithmetic such that for all except at most finitely many k∈N, φ(k)⇔k∈X. The simplest known such formula φ(x) satisfies {k∈N: φ(k)}=X. *** The set X={k∈N: (k>f(7)) ⇒ (f(7),k)∩P(n^2+1)≠∅} satisfies conditions (1)-(4). A more complicated set X⊆N satisfies conditions (1)-(5). No set X⊆N will satisfy conditions (1)-(4) forever, if for every algorithm with no inputs, 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 statement Φ implies that conditions (1)-(5) hold for X={1}∪P(n^2+1).
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