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Does there Exist an Algorithm which to Each Diophantine Equation Assigns an Integer which is Greater than the Modulus of Integer Solutions, if these Solutions form a Finite Set?

Tyszka A.

Fundamenta Informaticae

2013

Vol. 125, nr 1

85--99

Let En = {xi = 1; xi + xj = xk; xi · xj = xk : i; j; k ∈ {1,...,n}}. We conjecture that if a system S ⊆ En has only finitely many solutions in integers x1,...,xn, then each such solution (x1,...,xn) satisfies |x1|,...,|xn| ≤ 22n−1. Assuming the conjecture, we prove: (1) there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set, (2) if a set M Í \mathbbN is recursively enumerable but not recursive, then a finite-fold Diophantine representation of M does not exist.