Hilary Putnam. An unsolvable problem in number theory. The Journal of symbolic logic, vol. 25 no. 3 (for 1960, pub. 1962), pp. 220–232.

1972 ◽  
Vol 37 (3) ◽  
pp. 601-602
Author(s):  
Alonzo Church
Keyword(s):  
Number Theory ◽  
Hilary Putnam ◽  
Symbolic Logic ◽  
Unsolvable Problem

Martin Davis. On formally undecidable propositions of the Principia Mathematica and related systems. I. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, p. 4. - Kurt Gödel. On formally undecidable propositions of Principia Mathematica and related systems I. English translation of 4183 by Elliott Mendelson. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, pp. 5–38. - Martin Davis. On undecidable propositions of formal mathematical systems. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, pp. 39–40. - Kurt Gödel. On undecidable propositions of formal mathematical systems. A revised reprint of 41814. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, pp. 41–71. (See corrigenda, p. 74.) - Kurt Gödel. Postscriptum. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, pp. 71–73. - Kurt Gödel. On intuitionistic arithmetic and number theory. English translation of 41811 by Martin Davis. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, pp. 75–81. - Kurt Gödel. On the length of proofs. English translation of I 116 by Martin Davis. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, pp. 82–83. - Kurt Gödel. Remarks before the Princeton Bicentennial Conference on Problems in Mathematics — 1946. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, pp. 84–88. [Cf. XII 89(2).] - Martin Davis. An unsolvable problem of elementary number theory. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, p. 88. - Alonzo Church. An unsolvable problem of elementary number theory. A reprint of I 73. The undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions, edited by Martin Davis, Raven Press, Hewlett, New York, 1965, pp. 89–107.

1966 ◽  
Vol 31 (3) ◽  
pp. 484-494
Author(s):  
Stefan Bauer-Mengelberg
Keyword(s):  
New York ◽  
Number Theory ◽  
English Translation ◽  
Elementary Number Theory ◽  
Unsolvable Problem ◽  
Elementary Number ◽  
Kurt Gödel ◽  
Computable Functions ◽  
Martin Davis ◽  
Unsolvable Problems

An unsolvable problem in number theory

1960 ◽  
Vol 25 (3) ◽  
pp. 220-232 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Number Theory ◽  
Common Element ◽  
Rigorous Proof ◽  
Unsolvable Problem ◽  
Decision Method ◽  
Simple Characterization ◽  
The One

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.2In 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

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