Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
نویسندگان
چکیده
This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.
منابع مشابه
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 > ...
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