In 1970 Matiyasevich, building on earlier work of Davis–Putnam–Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert’s 10th problem is undecidable for Z. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extensions of number fields L/K, an elliptic curve E/K with rk(E(L))=rk(E(K))>0. This was done under BSD by Mazur and Rubin. In this talk I will explain joint work with Peter Koymans, where we use Green–Tao to construct the desired elliptic curves unconditionally, thus settling Hilbert 10 for every finitely generated infinite ring.