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Universal diophantine equation

Abstract

In 1961 Martin Davis, Hilary Putnam and Julia Robinson [2] proved that every recursively enumerable set W is exponential diophantine, i.e. can be represented in the form

Here P is a polynomial with integer coefficients and the variables range over positive integers.

In 1970 Ju. V. Matijasevič used this result to establish the unsolvability of Hilbert's tenth problem. Matijasevič proved [11] that the exponential relation y = 2x is diophantine This together with [2] implies that every recursively enumerable set is diophantine, i.e. every r.e. set Wcan be represented in the form

From this it follows that there does not exist an algorithm to decide solvability of diophantine equations. The nonexistence of such an algorithm follows immediately from the existence of r.e. nonrecursive sets.

Now it is well known that the recursively enumerable sets W1, W2, W3, … can be enumerated in such a way that the binary relation xWv is also recursively enumerable. Thus Matijasevič's theorem implies the existence of a diophantine equation U such that for all x and v,

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[1] A. Baker , Contributions to the theory of diophantine equations. I, Philosophical Transactions of the Royal Society of London, Series A, vol. 263 (1968), pp. 173191.

[2] Martin Davis , Hilary Putnam and Julia Robinson , The decision problem for exponential diophantine equations, Annals of Mathematics, vol. 74 (1961), pp. 425436 = Matematika, vol. 8 (5) (1964), pp. 69–79.

[4] Martin Davis , Hilbert's tenth problem is unsolvable, American Mathematical Monthly, vol. 80 (1973), pp. 233269.

[5] J.P. Jones , Undecidable diophantine equations, Bulletin of the American Mathematical Society (New Series), vol. 3, no. 2 (1980), pp. 859862.

[9] J.P. Jones , D. Sato , H. Wada and D. Wiens , Diophantine representation of the set of prime numbers, American Mathematical Monthly, vol. 83 (1976), pp. 449464. MR 54, 2615.

[12] Ju. V. Matijasevič , On recursive unsolvability of Hilbert's tenth problem, Proceedings of the IVth International Congress on Logic, Methodology and Philosophy of Science (Bucharest 1971), North-Holland, Amsterdam, 1973, pp. 89110. MR 57, 5711.

[19] Ju. V. Matijasevič , Some purely mathematical results inspired by mathematical logic, Foundations of mathematics and computabiliiy theory ( Butts and Hintakka , editors), D. Reidel Publishing Co., Dordrecht-Holland, 1977, pp. 121127.

[21] Julia Robinson , Hilbert's tenth problem, Proceedings of the Symposium on Pure Mathematics, vol. 20, American Mathematical Society, Providence, R.I., 1971, pp. 191194.

[25] Julia Robinson , Existential definability in arithmetic, Transactions of the American Mathematical Society, vol. 72 (1952), pp. 437449 = Matematika, vol. 8 (5) (1964), pp. 3–14. MR 14, 4.

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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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