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.
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