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
historicHistory & 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.
Connected problems
- inspired by The Fifth Busy Beaver Number BB(5)
- related Hilbert's Tenth Problem over ℚ
More in mathematical logic / number theory
- same area Hilbert's Tenth Problem over ℚ
References
- primaryRead the primary source →
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.
How to check it: the proof has four named layers — Davis (exponential Diophantine, 1950s), Putnam (1960), Robinson (the JR hypothesis), Matiyasevich (Fibonacci coding, 1970). The Matiyasevich finale is a few pages; the DPR machine it completes is the bulk. The live successor question — decidability over ℚ — has its own entry here.
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