References for Undecidability

 References:

The blog posts Diophantine set, Listable sets, Halting problem, DPRM Theorem are based on superb article by B Poonan. - Undecidability in Number Theory.
As the article points out, the undecidability problem was studied not only for Integers, but also for Rationals, Rings etc.,

One of the questions, Poonan posed earlier in the article - does $x^3+y^3+z^3=33$ has integer solutions? At the time of writing article answer to this question was unknown. The article prompted search for solution to above equation. In fact an integer solution was now found.