Are there a naturally defined set X⊆N and a constructively defined integer n such that (ω>card(X) ⇒ X⊆(-∞,n]) ∧ (the infiniteness of X is conjectured and cannot be decided by any known method) ∧ (a constructively defined algorithm decides X)?
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)). 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. 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 definition 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) X is naturally defined i.e. X has the simplest definition among known sets Y⊆N with the same set of known elements. *** The set X={k∈N: (k>f(7)) ⇒ (f(7),k)∩P(n^2+1)≠∅} satisfies conditions (1)-(4). 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=P(n^2+1).
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
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