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On the number of solutions of Diophantine equations


Author: Martin Davis
Journal: Proc. Amer. Math. Soc. 35 (1972), 552-554
MSC: Primary 10N05; Secondary 02E10
DOI: https://doi.org/10.1090/S0002-9939-1972-0304347-1
MathSciNet review: 0304347
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Abstract: For any nontrivial set of cardinal numbers $ \leqq {\aleph _0}$, it is shown that there is no algorithm for testing whether or not the number of positive integer solutions of a given polynomial Diophantine equation belongs to the set.


References [Enhancements On Off] (What's this?)

  • [1] Martin Davis, An explicit Diophantine definition of the exponential function, Comm. Pure Appl. Math. 24 (1971), 137-145. MR 42 #7632. MR 0272751 (42:7632)
  • [2] Ju. V. Matijasevič, Enumerable sets are Diophantine, Dokl. Akad. Nauk SSSR 191 (1970), 279-282=Soviet Math. Dokl. 11 (1970), 354-358.
  • [3] Julia Robinson, Hilbert's tenth problem, Proc. Sympos. Pure Math., vol. 20, Amer. Math. Soc., Providence, R.I., 1971, pp. 191-194. MR 0316234 (47:4782)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0304347-1
Keywords: Diophantine equations, unsolvable problems, Hilbert's tenth problem, recursive functions
Article copyright: © Copyright 1972 American Mathematical Society

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