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FIBONACCI NUMBERS AT MOST ONE AWAY FROM A PERFECT POWER YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract. The famous problem of determining all perfect powers in the Fi- bonacci 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. 1. Introduction We consider the Fibonacci sequence (Fn )n≥0 and the Lucas sequence (Ln )n≥0 both of which are solutions to the linear recurrence un+2 = un+1 + un , with the initial conditions F0 = 0, F1 = 1 and, respectively, L0 = 2, L1 = 1. The problem of determining all perfect powers in the Fibonacci sequence was a famous open problem for over 40 years, and has been resolved only recently [9]. Theorem 1. The only perfect powers among the Fibonacci numbers are F0 = 0, F1 = F2 = 1, F6 = 8 and F12 = 144. For the Lucas numbers, the only perfect powers are L1 = 1 and L3 = 4. Subsequent papers studied several multiplicative generalizations such as Fn = ayp (see [8]) and Fn1 · · · Fnr = yp with 1 ≤ r p (see [7]). Here, we consider t