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All functions \(g:\mathbb N\to\mathbb N\) which have a single-fold Diophantine representation are dominated by a limit-computable function \(f:\mathbb N\backslash\{0\}\to\mathbb N\) which is implemented in MuPAD and whose computability is an open problem. (English) Zbl 1349.03041

Daras, Nicholas J. (ed.) et al., Computation, cryptography, and network security. Cham: Springer (ISBN 978-3-319-18274-2/hbk; 978-3-319-18275-9/ebook). 577-590 (2015).
Summary: Let \(E_n=\{x_k=1,\,x_i+x_j=x_k,\, x_i\cdot x_j=x_k: i,j,k\in\{1,\dots,n\}\}\). For any integer \(n\geq 2214\), we define a system \(T E_n\) which has a unique integer solution \((a_1,\dots,a_n)\). We prove that the numbers \(a_1,\dots,a_n\) are positive and \(\max(a_1,\dots,a_n)>2^{2^n}\). For a positive integer \(n\), let \(f(n)\) denote the smallest non-negative integer \(b\) such that for each system \(S\subseteq E_n\) with a unique solution in non-negative integers \(x_1,\dots,x_n\), this solution belongs to \([0,b]^n\). We prove that if a function \(g:\mathbb N\to\mathbb N\) has a single-fold Diophantine representation, then \(f\) dominates \(g\). We present a MuPAD code which takes as input a positive integer \(n\), performs an infinite loop, returns a non-negative integer on each iteration, and returns \(f(n)\) on each sufficiently high iteration.
For the entire collection see [Zbl 1329.94003].

MSC:

03D20 Recursive functions and relations, subrecursive hierarchies
11U05 Decidability (number-theoretic aspects)

Software:

MuPAD
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