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On two conjectures regarding generalized sequence of derangements. (English) Zbl 1523.11048

Summary: In [Colloq. Math. 161, No. 1, 89–109 (2020; Zbl 1473.11036)] the third author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures stated in the paper cited above. The first conjecture regards the set of prime divisors of their terms. The latter one is devoted to the order of magnitude of considered sequences.

MSC:

11B83 Special sequences and polynomials
11B37 Recurrences

Citations:

Zbl 1473.11036
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References:

[1] J.-H. Evertse, On sums of S-units and linear recurrences, Compositio Mathematica 53 (1984), no. 2, 225-244. · Zbl 0547.10008
[2] P. Miska, On a generalization of the recurrence defining the number of derangements, Colloq. Math. 161 (2020), 89-109. · Zbl 1473.11036
[3] P. Miska, When a constant subsequence implies ultimate periodicity, Bulletin Of The Polish Academy Of Sciences Mathematics 67 (2019), no. 1, 41-52. · Zbl 1454.11043
[4] A.J. van der Poorten, H.P. Schlickewei, The growth conditions for recurrence sequences, Macquarie Math. Reports 82-0041 (1982).
[5] T.N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge U. Press, 1986. · Zbl 0606.10011
[6] K.-R. Yu, Report on \(p\)-adic logarithmic forms, A Panorama in Number Theory or The View from Baker’s Garden (2002), 11-25. · Zbl 1040.11055
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