Abstract

In this paper, we show that if ( u n ) n ⩾ 1 is a Lucas sequence, then the Diophantine equation u n · u n + 1 ⋯ · · u n + k = y m in integers n ⩾ 1 , k ⩾ 1 , m ⩾ 2 and y with | y | > 1 has only finitely many solutions. We also determine all such solutions when ( u n ) n ⩾ 1 is the sequence of Fibonacci numbers and when u n = ( x n - 1 ) / ( x - 1 ) for all n ⩾ 1 with some integer x > 1 .

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