Abstract
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.
Highlights
The first discussion on this project was possible when Japhet Odjoumani was at University of Graz under the support of Coimbra group
We demonstrate the method by applying it to the Tribonacci sequence T and the sequence L and prove the following two theorems: Theorem 2 The Diophantine equation Tn = px has at most one solution (n, x) with x = 0 unless p = 2
Note that the set S(U ) is finite and because C15 is effectively computable the set S(U ) is effectively computable. With this notation we immediately deduce that the Diophantine equation Un = px has at most one solution with x = 0 if p ∈/ S(U )
Summary
In the case of higher order recurrence sequences Shorey and Stewart [12] showed that under a dominant root assumption a solution (n, y, m) to Un = ym with y, m > 1 satisfies m < C, where C is an effectively computable constant. More precisely we show: Theorem 1 Let U = (Un)n≥0 be a simple, linear recurrence sequence of order at least two, satisfying the dominant root condition. In the case of the Tribonacci sequence and the Lucas sequence added by one the bound for n1 is small enough to find small lists of possible candidates of primes p such that the Diophantine equations Tn = px and Ln = px may have two solutions. Throughout the paper we will denote by C1, C2, . . . positive and effectively computable constants
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