Abstract
Let Bn = {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈ {1, . . . , n}} denote the system of equations in the variables x1, . . . , xn. For a positive integer n, let _(n) denote the smallest positive integer b such that for each system of equations S ⊆ Bn with a unique solution in positive integers x1, . . . , xn, this solution belongs to [1, b]n. Let g(1) = 1, and let g(n + 1) = 22g(n) for every positive integer n. We conjecture that ξ (n) 6 g(2n) for every positive integer n. We prove: (1) the function ξ : 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) < ξ (n) for every integer n > m; (3) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x1, . . . , xp) = 0 and returns a positive integer d with the following property: for every positive integers a1, . . . , ap, if the tuple (a1, . . . , ap) solely solves the equation D(x1, . . . , xp) = 0 in positive integers, then a1, . . . , ap 6 d; (4) the conjecture implies that if a set M ⊆ N has a single-fold Diophantine representation, then M is computable; (5) for every integer n > 9, the inequality ξ (n) < (22n−5 − 1)2n−5 + 1 implies that 22n−5 + 1 is composite.
References
[1] Davis M., On the number of solutions of Diophantine equations, Proc. Amer. Math. Soc. 35 (1972), no. 2, 552-554.Search in Google Scholar
[2] Davis M., Matiyasevich Yu., Robinson J., Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution; in: Mathematical developments arising from Hilbert problems (ed. F. E. Browder), Proc. Sympos. PureMath., vol. 28, Part 2, Amer. Math. Soc., Providence, RI, 1976, 323-378; reprinted in: The collected works of Julia Robinson (ed. S. Feferman), Amer. Math. Soc., Providence, RI, 1996, 269-324.Search in Google Scholar
[3] Křížek M., Luca F., Somer L., 17 lectures on Fermat numbers: from number theory to geometry, Springer, New York, 2001.10.1007/978-0-387-21850-2Search in Google Scholar
[4] Matiyasevich Yu., Hilbert’s tenth problem, MIT Press, Cambridge, MA, 1993.Search in Google Scholar
[5] Matiyasevich Yu., Hilbert’s tenth problem: what was done and what is to be done; in: Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Contemp. Math. 270, 1-47, Amer. Math. Soc., Providence, RI, 2000.Search in Google Scholar
[6] Matiyasevich Yu., Towards finite-fold Diophantine representations, J. Math. Sci. (N. Y.) vol. 171, no. 6, 2010, 745-752.10.1007/s10958-010-0179-4Search in Google Scholar
[7] Robinson J., Definability and decision problems in arithmetic, J. Symbolic Logic 14 (1949), no. 2, 98-114; reprinted in: The collected works of Julia Robinson (ed. S. Feferman), Amer. Math. Soc., Providence, RI, 1996, 7-23.Search in Google Scholar
[8] Skolem Th., Diophantische Gleichungen, Julius Springer, Berlin, 1938.Search in Google Scholar
[9] Smorynski C., A note on the number of zeros of polynomials and exponential polynomials, J. Symbolic Logic 42 (1977), no. 1, 99-106.Search in Google Scholar
[10] Smorynski C., Logical number theory, vol. I, Springer, Berlin, 1991.10.1007/978-3-642-75462-3Search in Google Scholar
[11] Tyszka A., A hypothetical way to compute an upper bound for the heights of solutions of a Diophantine equation with a finite number of solutions, Proceedings of the 2015 Federated Conference on Computer Science and Information Systems (eds. M. Ganzha, L. Maciaszek, M. Paprzycki); Annals of Computer Science and Information Systems, vol. 5, 709-716, IEEE Computer Society Press, 2015.10.15439/2015F41Search in Google Scholar
[12] Tyszka A., All functions g : N → N which have a singlefold Diophantine representation are dominated by a limitcomputable function f : N \ {0} → N which is implemented in MuPAD and whose computability is an open problem; in: Computation, cryptography, and network security (eds. N. J. Daras and M. Th. Rassias), Springer, Cham, Switzerland, 2015, 577-590.10.1007/978-3-319-18275-9_24Search in Google Scholar
[13] Tyszka A., Conjecturally computable functions which unconditionally do not have any finite-fold Diophantine representation, Inform. Process. Lett. 113 (2013), no. 19-21, 719-722.Search in Google Scholar
© 2017
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.
