The problem
Hilbert asked for an effective procedure which, given a polynomial equation with integer coefficients, decides whether it has an integer solution. Resolution: no such algorithm exists — integer-solvability of polynomials is undecidable.
Hilbert's Tenth Problem
historicReferences
History & significance
The path ran through America and Russia: Davis's 1950s hypotheses, Putnam's sharpening, Julia Robinson's crucial conditions (1961 with Davis and Putnam), and the decisive young Matiyasevich (1970) proving exponentiation is Diophantine via Fibonacci-number growth. MRDP: recursively enumerable sets ARE Diophantine sets — a perfect dictionary between computability theory and number theory.
The resolution (human proof)
Undecidability proven 1970 (Matiyasevich, completing Davis–Putnam–Robinson). The theorem's aftermath became a discipline: undecidability spreads to equations over rationals (open!), group theory, topology; negative solutions to Hilbert problems are rare and precious — this one reshaped three fields at once and made Julia Robinson a founding figure of mathematical logic.