ON A PROBLEM CONCERNING PERFECT POWERS IN LINEAR RECURRENCES Péter Kiss (EKTF, Hungary)

Abstract: For a linear recurrence sequence {Gn} n=0 of rational integers of order k ≥ 2 satisfying some conditions, we show that the equation sGrx = w q, where w> 1 and r are positive integers and s contains only given primes as its prime factors, implies the inequality q < q0, where q0 is an effective computable constant depending on the sequence, the prime factors of s and r. Let G = {Gn} n=0 be a linear recurrence sequence of order k ≥ 2 defined by Gn = A1Gn−1 +A2Gn−2 + · · ·+AkGn−k (n ≥ k), where A1,..., Ak are given rational integers with Ak 6 = 0 and the initial values G0, G1,..., Gk−1 are not all zero integers. We denote by α = α1, α2,..., αs the distinct roots of the polynomial g(x) = xk −A1x k−1 −A2x k−2 − · · · −Ak, furthermore we suppose that |α |> |αi | for 2 ≤ i ≤ s, and the roots α = α1, α2,..., αs have multiplicity m1 = 1,m2,...,ms. In this case |α |> 1 and, as it is well known, the terms of G can be writen in the form (1) Gn = aα n + g2(n)