Abstract
The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematics, combining the modular approach used in the proof of Fermat’s Last Theorem with Baker’s Theory. By contrast, using the knowledge of the all perfect powers in the Fibonacci and Lucas sequences, the determination of the perfect powers among the numbers Fn ± 1 is quite elementary.
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