Equations 𝑎𝑢^{𝑙}_{𝑛}=𝑏𝑢^{𝑘}ₘ satisfied by members of recurrence sequences

Schlickewei, H. P.; Schmidt, W. M.

doi:10.1090/S0002-9939-1993-1152290-4

Equations $au^ l_ n=bu^ k_ m$ satisfied by members of recurrence sequences
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by H. P. Schlickewei and W. M. Schmidt

Proc. Amer. Math. Soc. 118 (1993), 1043-1051

DOI: https://doi.org/10.1090/S0002-9939-1993-1152290-4

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Abstract:

Let ${\{ {u_n}\} _{n \in \mathbb {Z}}}$ be a linear recurrence sequence. Given $a \ne 0, b \ne 0$, and natural $k \ne l$, we study equations as indicated in the title in unknowns $n,m$. It turns out that under natural conditions on the sequence $\{ {u_n}\}$, there are only finitely many solutions.

References

Jan-Hendrik Evertse, On sums of $S$-units and linear recurrences, Compositio Math. 53 (1984), no. 2, 225–244. MR 766298
Michel Laurent, Équations exponentielles-polynômes et suites récurrentes linéaires. II, J. Number Theory 31 (1989), no. 1, 24–53 (French, with English summary). MR 978098, DOI 10.1016/0022-314X(89)90050-4

Aufgaben und Lehrsätze aus der Analysis

A. J. van der Poorten and H. P. Schlickewei, Zeros of recurrence sequences, Bull. Austral. Math. Soc. 44 (1991), no. 2, 215–223. MR 1126359, DOI 10.1017/S0004972700029646
Hans Peter Schlickewei, Multiplicities of algebraic linear recurrences, Acta Math. 170 (1993), no. 2, 151–180. MR 1226526, DOI 10.1007/BF02392784
H. P. Schlickewei and Wolfgang M. Schmidt, On polynomial-exponential equations, Math. Ann. 296 (1993), no. 2, 339–361. MR 1219906, DOI 10.1007/BF01445109

Linear equations in members of recurrence sequences

The intersection of recurrence sequences