A common approach to Brocard's problem, Landau's problem, and the twin prime problem

Let f(1)=2, f(2)=4, f(n+1)=f(n)! for n>1. E.Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. This conjecture implies the following unproven statement F: card(P(n^2+1)) P(n^2+1) subset [2,f(7)]. Let B denote the system: {x_i!=x_k: i,k in {1,...,9}} cup {x_i cdot x_j=x_k: i,j,k in {1,...,9}}. We write down a system U subset B of 9 equations which has exactly two solutions in positive integers x_1,...,x_9: (1,...,1) and (f(1),...,f(9)). We write down a system A subset B of 8 equations. Let L 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 leq f(9). The statement L is equivalent to the statement F. This heuristically proves the statement F. This proof does not yield that card(P(n^2+1))=omega. 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 subset 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)=omega. (3) A known algorithm for every k in N decides whether or not k in X. (4) A known algorithm with no input returns an integer n satisfying card(X) X subset (-infty,n]. (5) X has the simplest definition among known sets Y subset N with the same set of known elements. *** The set X={k in N: (f(7) (f(7),k) cap P(n^2+1) neq emptyset} satisfies conditions (1)-(4). No set X subset N will satisfy conditions (1)-(4) 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 statement F implies that conditions (1)-(5) hold for X=P(n^2+1).