Hilbert's 10th Problem Extended: Undecidability Proved for Broader Rings

Mathematicians have solved a major extension of Hilbert's 10th problem, proving that determining whether Diophantine equations have solutions is undecidable for a vast class of number rings. Building on Yuri Matiyasevich's 1970 proof for integer solutions, the work utilizes elliptic curves and quadratic twists to overcome limitations of previous approaches with non-integer solutions. This breakthrough not only deepens our understanding of the limits of computability but also provides new tools for mathematical research.