Gary Walsh: Lucas sequences with rational roots

We discuss the problem of determining the arithmetic structure of terms in binary linear recurrences. In particular, in the case that the sequence is a Lucas sequence of the second kind, whose roots of the companion polynomial are rational, Henri Darmon and Loïc Merel, with the help of Bjorn Poonen, proved that no squares exist in the sequence beyond terms of index 3. We discuss an analogous result for Lucas sequences with rational roots, but for sequences of the first kind. The results we discuss are obtained by applying recent Diophantine applications of the modularity of elliptic curves by Michael Bennett and Christopher Skinner, Ken Ribet, Richard Taylor and Andrew Wiles, and also recent improvements to effective Chabauty methods developed by Nils Bruin.

Bibliography

1

Michael A. Bennett and Chris M. Skinner.
Ternary Diophantine equations via Galois representations and modular forms.
Canad. J. Math., 56(1):23-54, 2004.

2

N. R. Bruin.
Chabauty methods and covering techniques applied to generalized Fermat equations, volume 133 of CWI Tract.
Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 2002.
Dissertation, University of Leiden, Leiden, 1999.

3

Henri Darmon and Loïc Merel.
Winding quotients and some variants of Fermat's last theorem.
J. Reine Angew. Math., 490:81-100, 1997.

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