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THE EXPONENTIAL DIOPHANTINE PROBLEM FOR ${\mathbb {Q}}$

Part of: Diophantine equations Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  21 July 2020

MIHAI PRUNESCU
MIHAI PRUNESCU*
Affiliation:
FACULTY OF MATHEMATICS AND INFORMATICS UNIVERSITY OF BUCHAREST, BUCHAREST, ROMANIA and SIMION STOILOW INSTITUTE OF MATHEMATICS OF THE ROMANIAN ACADEMY RESEARCH UNIT 5, P.O. BOX 1–764 RO-014700BUCHAREST, ROMANIAE-mail: [email protected]: [email protected]
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Abstract

We show that the set of natural numbers has an exponential diophantine definition in the rationals. It follows that the corresponding decision problem is undecidable.

Keywords

exponential diophantine equationsrational numbersdecision problemsundecidability

MSC classification

Primary: 11D61: Exponential equations
Secondary: 14G05: Rational points
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 671 - 672
Copyright
© The Association for Symbolic Logic 2020

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References

REFERENCES

Dickson, L. E., History of the Theory of Numbers, vol. 2, Carnegie Institute, Washington, 1919; reprinted by Chelsea, New York, 1966.Google Scholar
Davis, M., Putnam, H. and Robinson, J., The decision problem for exponential diophantine equations . Annals of Mathematics, vol. 74 (1961), no. 3, pp. 425–436.CrossRefGoogle Scholar
Matiyasevich, Y., Hilbert’s Tenth Problem, MIT Press, Cambridge, MA, 1993.Google Scholar