A conjecture on integer arithmetic which implies that 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

Abstract

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 integers x 1 ,. .. , x n , then each such solution (x 1 ,. .. , x n) satisfies max |x 1 |,. .. , |x n | ≤ 2 2 n−1. The conjecture implies that 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. We describe an algorithm whose execution never terminates. If the conjecture is true, then the algorithm sequentially displays all integers n ≥ 2. If the conjecture is false, then the algorithm has a finite output which ends with an integer tuple (a 1 ,. .. , a n), where n ≥ 4, 2 2 n−1 < max |a 1 |,. .. , |a n | ≤ 2 2 n , and the system {x i + x j = x k : (i, j, k ∈ {1,. .. , n}) ∧ (a i + a j = a k)} ∪ {x i • x j = x k : (i, j, k ∈ {1,. .. , n}) ∧ (a i • a j = a k)} is a counterexample to the conjecture. The algorithm is implemented in MuPAD and Pascal.

All Functions $$g: \mathbb{N} \rightarrow \mathbb{N}$$ Which have a Single-Fold Diophantine Representation are Dominated by a Limit-Computable Function $$f: \mathbb{N}\setminus \{0\} \rightarrow \mathbb{N}$$ Which is Implemented in MuPAD and Whose Computability is an Open Problem

“…, x n . Theorem 1 disproves the conjecture in [11], where the author proposed the upper bound 2 2 n−1 for modulus of integer solutions to any system S ⊆ E n which has only finitely many solutions in integers x 1 , . .…”

A new characterization of computable functions

“…For K ∈ Rng, the Lemma is proved in [8]. For concrete Diophantine equations, it is possible to find much smaller equivalent systems of equations of the forms x i = 1,…”