Information Processing Letters 113 (2013) 719–722 Contents lists available at SciVerse ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl Conjecturally computable functions which unconditionally do not have any finite-fold Diophantine representation Apoloniusz Tyszka University of Agriculture, Faculty of Production and Power Engineering, Balicka 116B, 30-149 Kraków, Poland article info abstract Article history: Received 9 January 2013 Received in revised form 1 July 2013 Accepted 4 July 2013 Available online 11 July 2013 Communicated by J. Torán Keywords: Theory of computation Davis–Putnam–Robinson–Matiyasevich theorem Diophantine equation with a finite number of solutions Matiyasevich’s conjecture on finite-fold Diophantine representations We define functions f 1 , f 2 , g 1 , g 2 : N \{0}→ N \{0} and prove that they do not have any finite-fold Diophantine representation. We conjecture that if a system S ⊆{x i + x j = x k , x i · x j = x k : i , j , k ∈{1,..., n}} has only finitely many solutions in positive integers x 1 ,..., x n , then each such solution (x 1 ,..., x n ) satisfies x 1 ,..., x n  2 2 n−1 . Assuming the conjecture, we prove: (1) the functions f 1 , f 2 , g 1 , g 2 are computable, (2) there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions if the solution set is finite, (3) a finite-fold Diophantine representation of the function N ∋ n → 2 n ∈ N does not exist, (4) if a set M ⊆ N is recursively enumerable but not recursive, then a finite-fold Diophantine representation of M does not exist.  2013 Elsevier B.V. All rights reserved. The negative solution to the Hilbert’s 10th Problem is that there is no algorithm for determining whether a Dio- phantine equation has an integer solution. This was proved by Yuri Matiyasevich in 1970. There is also no algorithm determining whether the number of positive integer solu- tions to a Diophantine equation is finite or infinite, see [2]. Let us pose the following problem. Problem. Is there an algorithm which takes as input a Dio- phantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions if the solution set is fi- nite? We remind the reader that the height of a rational number p q is defined by max(| p|, |q|) provided p q is written in lowest terms. In this note, we discuss a conjecture which implies a positive answer to all versions of the Problem. E-mail address: rttyszka@cyf-kr.edu.pl. Conjecture. If a system S ⊆ E n =  x i = 1, x i + x j = x k , x i · x j = x k : i , j , k ∈{1,..., n}  has only finitely many solutions in positive integers x 1 ,..., x n , then each such solution (x 1 ,..., x n ) satisfies x 1 ,..., x n  2 2 n−1 . The analogous conjecture for integer solutions is dis- cussed in [1,12,16]. The system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x 1 + x 1 = x 2 x 1 · x 1 = x 2 x 2 · x 2 = x 3 x 3 · x 3 = x 4 ··· x n−1 · x n−1 = x n has exactly two integer solutions, namely (0,..., 0) and (2, 4, 16, 256,..., 2 2 n−2 , 2 2 n−1 ). Therefore, the bound 2 2 n−1 is tight for any n  2. In the domain of positive integers, the equation x = 1 is equivalent to the equation x · x = x. Therefore, without 0020-0190/$ – see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ipl.2013.07.004