
Let E_n be the set of all equations of the form x_i = 1, or x_i + x_j = x_k, or x_i * x_j = x_k, where i,j,k range over {1,...,n}. Moreover let K be one of the rings Z,Q,R,C. We construct a system S of equations from E_{21} such that S has infinitely many integer solutions and S has no integer solution in the cube [2^{2^{211}},2^{2^{211}}]^{21}. We conjecture that if a system S, contained in E_n, has a finite number of solutions in K, then each such solution (x_1,...,x_n) satisfies (x_1,...,x_n) \in [0,2^{2^{n1}}]^n. Applying this conjecture for K=Z, we prove that if a Diophantine equation has only finitely many integer (rational) solutions, then these solutions can be algorithmically found. On the other hand, an affirmative answer to the famous open problem whether each listable subset M of Z^n has a finitefold Diophantine representation would falsify our conjecture for K=Z. Full text: http://arxiv.org/abs/0901.2093 References: A. Kozlowski and A. Tyszka, A Conjecture of Apoloniusz Tyszka on the Addition of Rational Numbers, 2008, Yu. Matiyasevich, Hilbert's tenth problem: what was done and what is to be done. Hilbert's tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), 147, Contemp. Math. 270, Amer. Math. Soc., Providence, RI, 2000 A. Tyszka, A system of equations, SIAM Problems and Solutions (electronic only), Problem 07006, 2007, A. Tyszka, Some conjectures on addition and multiplication of complex (real) numbers, Int. Math. Forum 4 (2009), no. 912, 521530, A. Tyszka, Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients 