On prime powers in linear recurrence sequences.

In this paper we consider the Diophantine equation U n = p x where U n is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on U n , we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution ( n ,  x ) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.