Small Prime Powers in the Fibonacci Sequence

Abstract

It is shown that there are no non-trivial fifth-, seventh-, eleventh-, thirteenth- or seventeenth powers in the Fibonacci sequence. For eleventh, thirteenth- and seventeenth powers an alternative (to the usual exhaustive check of products of powers of fundamental units) method is used to overcome the problem of having a large number of independent units and relatively high bounds on their exponents. It is envisaged that the same method can be used to decide the question of the existence of higher small prime powers in the Fibonacci sequence and that the method can be applied to other binary recurrence sequences. The alternative method mentioned may have wider applications.