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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 0272751, https://doi.org/10.1002/cpa.3160240205
  • [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, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 191–194. MR 0316234

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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