Title: An algebraic proof that an arithmetically defined function f:N\{0}-->N\{0} does not have any Diophantine representation
Abstract: Let B_n={x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let \xi(n) denote the smallest positive integer b such that for each system S \subseteq B_n with a unique solution in positive integers x_1,...,x_n, this solution belongs to [1,b]^n. Let g(1)=1, and let g(n+1)=2^{2^{g(n)}} for every positive integer n. We conjecture that \xi(n) \leq g(2n) for every positive integer n. We prove: (1) the function \xi: N\{0}-->N\{0} is computable in the limit; (2) if a function f:N\{0}-->N\{0} has a single-fold Diophantine representation, then there exists a positive integer m such that f(n) m; (3) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x_1,...,x_p)=0 and returns a positive integer d with the following property: for every positive integers a_1,...,a_p, if the tuple (a_1,...,a_p) solely solves the equation D(x_1,...,x_p)=0 in positive integers, then a_1,...,a_p \leq d; (4) the conjecture implies that if a set M \subseteq N has a single-fold Diophantine representation, then M is computable; (5) for every integer n>9, the inequality \xi(n)<(2^{2^{n-5}}-1)^{2^{n-5}}+1 implies that 2^{2^{n-5}}+1 is composite.
Publication Year: 2014
Publication Date: 2014-04-24
Language: en
Type: article
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