Abstract: In 1900, in the second ICM, Hilbert proposed a list of 23 problems that shaped much of mathematics throughout the past century. The tenth problem asked for an algorithm capable of deciding the solvability over the integer of any polynomial equation, in any number of variables. In 1970 Matiyasevich, building on earlier work of Robinson--Davis--Putnam, managed to prove that no such algorithm exists. Soon after the effort of researchers shifted on establishing the same undecidability result for more general number systems. Thanks to the effort of a number of mathematicians, the problem has been eventually reduced to the task of producing "diophantine stable" elliptic curves. In 2009, Mazur--Rubin solved this conditionally on BSD. I will present joint work with Peter Koymans where we establish the same result unconditionally, combining 2-descent with additive combinatorics. As a consequence of our work, Hilbert 10th problem has a negative answer for any finitely generated infinite commutative ring.