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

The k-generalized Fibonacci sequence (Fn(k))n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k >= 2, is defined by the values 0,0, horizontal ellipsis ,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 Fm(k)=mt, with t>1 and m>k+1, has only solutions F-12((2))=122 and F-9(3)=9(2).