New Proofs Expand the Limits of What Cannot Be Known

A significant mathematical problem extending Hilbert's 10th problem has been solved. Two independent teams proved that for many number systems beyond integers, no general algorithm exists to determine if Diophantine equations have solutions. This expands on Matiyasevich's 1970 proof for integers. The new proofs utilize complex techniques involving elliptic curves and prime numbers. This result highlights fundamental limits to mathematical knowledge and computation, revealing that some problems in basic math remain unsolvable.