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Hilbert’s Tenth Problem

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An Invitation to Mathematical Logic

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 301))

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Abstract

We discus parts of the negative solution to Hilbert’s tenth Problem, showing that there is no algorithm to decide if a Diophantine equation has an integer solution.

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Notes

  1. 1.

    Kreisel, in his review [55] of [14], expressed skepticism that one could reduce all questions about Diophantine equations to ones of a bounded degree and thought this was evidence their program would fail.

  2. 2.

    Our usual pairing function \(\pi (x,y)=2^x(2y+1)-1\) is also Diophantine. But it is not obvious that this is true until we know \(y=2^x\) is Diophantine.

References

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    David Marker

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  1. David Marker
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© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

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Marker, D. (2024). Hilbert’s Tenth Problem. In: An Invitation to Mathematical Logic. Graduate Texts in Mathematics, vol 301. Springer, Cham. https://doi.org/10.1007/978-3-031-55368-4_14

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