Undecidability in number theory

TL;DR

This paper discusses classical and recent undecidability results in number theory, focusing on Hilbert's 10th problem and the definability of integers within rationals, highlighting significant theoretical advances.

Contribution

It presents a comprehensive overview of undecidability in number theory and introduces a new result on the universal definability of integers in the rationals.

Findings

Hilbert's 10th problem is undecidable for integers.

The authors sketch a proof that integers are universally definable in the rationals.

Recent developments extend undecidability results to rings beyond integers.

Abstract

These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers are universally definable in the rationals.