Diophantine equations with products of consecutive terms in Lucas sequences

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