Coincidences in generalized Fibonacci sequences

Abstract

The k -generalized Fibonacci sequence ( F n ( k ) ) n resembles the Fibonacci sequence in that it starts with 0 , … , 0 , 1 ( k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we are interested in finding the integers that appear in different generalized Fibonacci sequences; i.e., we study the Diophantine equation F n ( k ) = F m ( ℓ ) in positive integers n , k , m , ℓ with k , ℓ ⩾ 2 .