Two hypothetical properties of integer arithmetic and their consequences for Diophantine problems
Apoloniusz Tyszka · 2009
Let B_n(Z)={(x_1,...,x_n) \in Z^n: \exists y_1,...,y_n \in Z (|x_1| y_i+y_j=y_k)) & \forall i,j,k \in {1,...,n} (x_i*x_j=x_k => y_i*y_j=y_k)}. We conjecture: (1) for each integers x_1,...,x_n, if 2^(2^(n-1))t(M), then M is infinite. By Conjecture (1), if a Diophantine equation has only finitely many solutions in integers (non-negative integers, rationals), then their heights are bounded from above by a computable function of the degree and the coefficients of the equation. Conjecture (1) implies that the set of Diophantine equations which have infinitely many solutions in integers (non-negative integers) is recursively enumerable. Conjecture (1) stated for any computable bound instead of 2^(2^(n-1)) remains in contradiction to Matiyasevich's conjecture that each recursively enumerable set M \subseteq N^n has a finite-fold Diophantine representation. Conjecture (2) implies the well-known existence of a Diophantine equation whose solvability in integers is logically undecidable.