Almost powers in the Lucas sequence

The famous problem of determining all perfect powers in the Fibonacci sequence (Fn)n≥0 and in the Lucas sequence (Ln)n≥0 has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations Ln = qy, with a > 0 and p ≥ 2, for all primes q < 1087 and indeed for all but 13 primes q < 10. Here the strategy of [10] is not sufficient due to the sizes of the bounds and complicated nature of the Thue equations involved. The novelty in the present paper is the use of the double-Frey approach to simplify the Thue equations and to cope with the large bounds obtained from Baker’s theory. Manuscrit reçu le 20 octobre 2007. F. Luca is supported by grants SEP-CONACyT 79685 and PAPIIT 100508. S. Siksek is supported by a grant from the UK Engineering and Physical Sciences Research Council, and by a Marie–Curie International Reintegration Grant. 556 Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek