By Idun Reiten, Sverre O. Smalø, Øyvind Solberg

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**Example text**

General theorems for powers The general theorem of Shorey (1981, 1983) (valid for all nondegenerate binary recurrence sequences) was proved using sharp lower bounds for linear forms in logarithms by Baker (1973), plus a p-adic version by van der Poorten (1977), assisted by another result of Kotov (1976). A result of Shorey (1977) may also be used, as suggested by ¨. 1) Let (P, Q) ∈ S, k ≥ 1. There exists an eﬀectively computable number C = C(P, Q, k) > 0 such that if n ≥ 1, |x| ≥ 2, h ≥ 2 and 30 1.

36:721–736. 1969 H. London and R. Finkelstein (alias R. Steiner). On Fibonacci and Lucas numbers which are perfect powers. , 7:476-481 and 487. 1972 J. H. E. Cohn. Squares in some recurrence sequences. Paciﬁc J. , 41:631–646. n −1 1972 K. Inkeri. On the Diophantic equation a xx−1 = y m . , 21:299–311. 1973 A. Baker. A sharpening for the bounds of linear forms in logarithms, II. , 24:33–36. 1973 H. London and R. Finkelstein (alias R. Steiner). Mordell’s Equation y 2 − k = x3 . Bowling Green State University Press, Bowling Green, OH.

Assume that P , Q are odd. 1. (a) If 1 < m < n and Um Un = ✷, then (m, n) ∈ {(2, 3), (2, 12), (3, 6), (5, 10)} or n = 3m, (b) If 1 < m, Um U3m = ✷, then m is odd, 3 m, Q ≡ 1 (mod 4), −Q = +1, and P < |Q + 1|. P (c) If P , m > 1 are given, there exists an eﬀectively computable constant C > 0 such that if Q is as in the hypotheses, and if Um U3m = ✷, then |Q| < C. (d) If P , Q are given as above, there exists an eﬀectively computable C > 0 such that if m > 1 and Um U3m = ✷, then m < C. 2. (a) If 1 < m < n and Vm Vn = ✷, then n = 3m.