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An unsolvable problem in number theory. (English) Zbl 0108.00701

By using Davis’ representation of recursively enumerable predicates [ Zbl 0051.245*)] the author shows that the following and related problems are (recursively) unsolvable: to decide, for any given polynomial with integral coefficients, whether it represents every integer. This is probably the “simplest” problem of number theory, formulated in language familiar to the “ordinary” number theoretician, that has so far been shown to be undecidable.
Also the author straightens out related parts of Myhill’s paper [ Zbl 0052.249*)], by (a) giving a full proof of Myhill’s result: There is no recursive method for telling, for any polynomial \(P\), whether \((\exists\;y)\) \((x_1\leq < y, \ldots, x_n\leq y)\) \([P (y, x_1, \ldots, x_n) = 0]\), and (b) refuting Myhill’s claim that there exists a fixed \(P_0\), containing an additional variable \(a\), with the following property: there is no recursive method for telling for any given \(a\), whether \[ (\exists\;y)\quad (x_1\leq < y, \ldots, x_n\leq y)\quad [P_0(y, x_1, \ldots, x_n,a) = 0].\tag{*} \] For he shows that, for any \(P\), either (*) is true for no value of \(a\) or only for finitely many.
Reviewer: G. Kreisel

MSC:

03-XX Mathematical logic and foundations
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[1] this Journal 23 pp 183– (1958)
[2] Actes du Xième Congrès International de Philosophie (Brussels, August 20–26, 1953), vol. XIV, Volume complémentaire et communications du Colloque de Logique pp 50– (1953)
[3] Sentences undecidable in formalized arithmetic pp 117– (1952)
[4] this Journal 18 pp 33– (1953)
[5] DOI: 10.1090/S0002-9947-1952-0048374-2
[6] Computability and unsolvability pp 210– (1958)
[7] this Journal 21 pp 162– (1956)
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