×

Lehmer sequences in finite groups. (English) Zbl 1492.11020

Ukr. Math. J. 68, No. 2, 193-202 (2016) and Ukr. Mat. Zh. 68, No. 2, 175-182 (2016).
Summary: We study the Lehmer sequences modulo \(m\). Moreover, we define the Lehmer orbit and the basic Lehmer orbit of a 2-generator group \(G\) for a generating pair \((x, y) \in G\) and examine the lengths of the periods of these orbits. Furthermore, we obtain the Lehmer lengths and the basic Lehmer lengths of the Fox groups \(G_{1,t}\) for \(t \geq 3\).

MSC:

11B37 Recurrences
20F05 Generators, relations, and presentations of groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. M. Campbell, H. Doostie, and E. F. Robertson, “Fibonacci length of generating pairs in groups,” Appl. Fibonacci Numbers, 3, 27-35 (1990). · Zbl 0741.20025 · doi:10.1007/978-94-009-1910-5_4
[2] C. M. Campbell, P. P. Campbell, H. Doostie, and E. F. Robertson, “Fibonacci lengths for certain metacyclic groups,” Algebra Colloq., 11, No. 2, 215-229 (2004). · Zbl 1069.20021
[3] O. Deveci, “The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite groups,” Util. Math., 98, 257-270 (2015). · Zbl 1369.11011
[4] O. Deveci and E. Karaduman, “The Pell sequences in finite groups,” Util. Math., 96, 263-276 (2015). · Zbl 1378.11025
[5] A. T. Fuller, “The period of pseudo-random numbers generated Lehmer’s congruential method,” Comput. J., 19, No. 2, 173-177 (1976). · Zbl 0321.65002 · doi:10.1093/comjnl/19.2.173
[6] D. H. Lehmer, “An extended theory of Lucas functions,” Ann. Math., 31, No. 2, 419-448 (1930). · JFM 56.0874.04 · doi:10.2307/1968235
[7] B. M. Phong, “On generalized Lehmer sequences,” Acta Math. Hung., 57, No. 3-4, 201-211 (1991). · Zbl 0632.10005 · doi:10.1007/BF01903671
[8] A. Rotkiewicz, “On strong Lehmer pseuduprimes in the case of negative discriminant in arithmetic progressions,” Acta Arithm., 68, No. 2, 145-151 (1994). · Zbl 0822.11016
[9] D. D. Wall, “Fibonacci series modulo <Emphasis Type=”Italic“>m,” Amer. Math. Mon., 67, 525-532 (1960). · Zbl 0101.03201 · doi:10.2307/2309169
[10] H. J. Wilcox, “Fibonacci sequences of period <Emphasis Type=”Italic“>n in groups,” Fibonacci Quart., 24, 356-361 (1986). · Zbl 0603.10011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.