Let R be an infinite ring that is finitely generated over Z. Hilbert’s 10th problem for R asks: does there exist an algorithm that given as input a polynomial f in R[X_1, …, X_n] outputs yes if f has a zero in R and no otherwise. Mazur and Rubin proved that Hilbert’s 10th problem is undecidable assuming finiteness of Sha. I will outline how to prove the same result without assuming finiteness of Sha. The key new input is a recent version of the Green—Tao theorem for number fields. This is joint work with Carlo Pagano.