Hilbert's Tenth Problem - History

History

Year Events
1944 Emil Leon Post declares that Hilbert's tenth problem "begs for an unsolvability proof".
1949 Martin Davis uses Kurt Gödel's method for applying the Chinese Remainder Theorem as a coding trick to obtain his normal form for recursively enumerable sets:

where is a polynomial with integer coefficients. Purely formally, it is only the bounded universal quantifier that stands in the way of this being a Diophantine definition.

Using a non-constructive but easy proof, he notes that there is a Diophantine set whose complement is not Diophantine. Because the recursively enumerable sets also are not closed under complementation, he conjectures that the two classes are identical.
1950 Julia Robinson, unaware of Davis's work, but grasping the key importance of the exponential function, attempts to prove that EXP, the set of triplets
for which

is Diophantine. Not succeeding, she is led to her hypothesis (later called J.R.): There is a Diophantine set of pairs such that

but for every ,

such that

Using properties of the Pell equation, she proves that J.R. implies that EXP is Diophantine. Finally she shows, that if EXP is Diophantine so are the binomial coefficients, the factorial, and the primes.
1959 Working together, Davis and Putnam study exponential Diophantine sets: sets definable by Diophantine equations in which some of the exponents may be unknowns. Using the Davis normal form together with Robinson's methods, but assuming the then unproved conjecture that there are arbitrarily long arithmetic progressions consisting of prime numbers, they prove that every recursively enumerable set is exponential Diophantine, and as a consequence, that J.R. implies that every recursively enumerable set is Diophantine, and that Hilbert's tenth problem is unsolvable.
1960 Robinson shows how to avoid the unproved conjecture about primes in arithmetic progression and then greatly simplifies the proof. J.R. is revealed as the key to further progress, though many doubt that it is true.
1961–1969 Davis and Putnam find various propositions that imply J.R. Yuri Matiyasevich publishes some reductions of Hilbert's tenth problem. Robinson shows that the existence of an infinite Diophantine set of primes would suffice to establish J.R.
1970 Matiyasevich exhibits a system of 10 simultaneous first and second degree equations which provide a Diophantine definition of the set of pairs such that
where is the nth Fibonacci number.

This proves J.R. and thus completes the proof that all recursively enumerable sets are Diophantine, and that therefore Hilbert's tenth problem is unsolvable.

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