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An unsolvable problem in number theory

Published online by Cambridge University Press:  12 March 2014

Hilary Putnam
Hilary Putnam*
Affiliation:
Princeton University
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Extract

This paper presents several results whose common element is their connection with Hubert's tenth problem.1 The one that seems most striking (perhaps because it can be stated with a minimum of technical apparatus) is this: there does not exist an algorithm for determining whether or not a polynomial (in n variables) represents every integer.2

In addition, the present paper (i) gives a simple characterization of the diophantine sets (of positive, non-negative, and arbitrary integers), and (ii) gives a rigorous proof of Myhill's theorem [6] that there is no decision method for statements of the form

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 25 , Issue 3 , September 1960 , pp. 220 - 232
Copyright
Copyright © Association for Symbolic Logic 1960

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References

[1]Davis, Martin, Arithmetical problems and recursively enumerable predicates, this Journal, vol. 18 (1953), pp. 33–41.Google Scholar
[2]Davis, Martin, Computability and unsolvability, New York, Toronto, and London (McGraw-Hill), 1958, xxv + 210 pp.Google Scholar
[3]Robinson, Julia, Existential definability in arithmetic, Transactions of the American Mathematical Society, vol. 72 (1952), pp. 437–449.CrossRefGoogle Scholar
[4]Robinson, Raphael, Arithmetical representation of recursively enumerable sets, this Journal, vol. 21 (1956), pp. 162–186.Google Scholar
[5]Mostowski, Andrzej, Sentences undecidable in formalized arithmetic, Amsterdam (North Holland), 1952, VIII + 117 pp.Google Scholar
[6]Myhill, John, Three contributions to recursive function theory, Actes du Xième Congrès International de Philosophie (Brussels, August 20–26, 1953), vol. XIV, Volume complémentaire et communications du Colloque de Logique, Amsterdam (North Holland), 1953, pp. 50–59.Google Scholar
[7]Davis, Martin and Putnam, Hilary, Reductions of Hubert's tenth problem, this Journal, vol. 23 (1958), pp. 183–187.Google Scholar