117b – Undecidability and Incompleteness – Lecture 2

The core of the proof of the undecidability of the tenth problem is the proof that exponentiation is Diophantine. We reduce this to proving that certain sequences given by second-order recurrence equations are Diophantine. These are the Matiyasevich sequences: For b³4,

ab(n+2)=bab(n+1)-ab(n)  for all n.

We begin the proof that they are Diophantine by analyzing some of the algebraic identities their terms satisfy.

Additional reference:

  • Unsolvable problems, by M. Davis. In Handbook of mathematical logic, J. Barwise, ed., North-Holland (1977), 567-594.


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