On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes

Pavel Trojovský

doi:10.3390/math7080700

Abstract

The k-generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the Diophantine equation F m ( k ) = m t , with t > 1 and m > k + 1 , has only solutions F 12 ( 2 ) = 12 2 and F 9 ( 3 ) = 9 2 .

Highlights

The well-known Fibonacci sequence ( Fn )n≥0 is given by the following recurrence of the second orderFn+2 = Fn+1 + Fn, for n ≥ 0, with the initial terms F0 = 0 and F1 = 1
One of the famous classical problems, which has attracted a attention of many mathematicians during the last thirty years of the twenty century, was the problem of finding perfect powers in the sequence of Fibonacci numbers
Chaves and Marques [10] proved that the Diophantine equation ( Fn )2 + ( Fn+1 )2 = Fm has no solution in positive integers m, n, k for n > 1 and k ≥ 3

Summary

Introduction

The well-known Fibonacci sequence ( Fn )n≥0 is given by the following recurrence of the second order. In 2006 Bugeaud et al [2] (Theorem 1), confirmed these expectations, as they showed that 0, 1, 8 and 144 are the only perfect powers in the sequence of Fibonacci numbers. The result itself is extremely interesting, but the way of its proof is even more interesting to mathematicians, as this proof combined two powerful techniques from number theory, namely, Baker’s theory on linear forms in logarithms and the tools from the Wiles’s proof of the Last Fermat Theorem This result started great efforts for finding perfect powers in some generalized Fibonacci sequences. Chaves and Marques [10] proved that the Diophantine equation ( Fn )2 + ( Fn+1 )2 = Fm has no solution in positive integers m, n, k for n > 1 and k ≥ 3 This result was generalized by Bednařík et al [11],. Our main approach of the proof of Theorem 1 is a similar as in [14], as we think that this kind of approach is very helpful to the readers

Auxiliary Results

Upper Bounds for m in Terms of t and for t in Terms of k and m

An Upper Bound for k in Terms of log m and t in Terms of m

The Proof of Theorem 1

Conclusions