The Complex-type Pell p-Numbers in Finite Groups

Year 2021, Volume: 6 Issue: 3, 142 - 147, 30.12.2021

Yeşim Aküzüm

Abstract

In this paper, we study the complex-type Pell p-numbers modulo m and then we obtain the periods and the ranks of the complex type Pell p-numbers modulo m. Also, we give some results on the periods and the ranks of the complex-type Pell p-numbers modulo m. Then, we consider the multiplicative orders of the complex-type Pell p-matrix when read modulo m. Also, we give the relationships between the periods of the complex-type Pell p-numbers modulo m and the orders of the cyclic groups produced. We redefine the complex-type Pell p-numbers by means of the elements of groups. Finally, we obtain the periods of the complex-type Pell 2-numbers in the semidihedral group SD_2^m , (m ≥ 4) as applications of the results.

Keywords

The complex-type Pell p-numbers, Matrix, Representation, Group, Period, Rank

References

• [1] Adigüzel Z., Erdag, O., Deveci O. Padovan-circulant-Hurwitz Dizilerinin m Modülüne Göre Periyotları. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 12(2), 2019, 783-787.
• [2] Akuzum Y., Aydin H., Deveci O. The Complex–Type Pell p-Numbers. submitted.
• [3] Akuzum Y. Deveci O. The Hadamard-type k-step Fibonacci sequences in groups. Commun. Algebra. 48(7), 2020, 2844-2856.
• [4] Akuzum Y., Deveci O. On The Jacobsthal-Padovan p-Sequences in Groups. Topological Algebra its Appl. 5 (1), 2017, 63-66.
• [5] Campbell C.M., Campbell P.P. The Fibonacci lengths of binary polyhedral groups and related groups. Congressus Numerantium. 194, 2009, 95–102.
• [6] Deveci O., Akdeniz M., Akuzum Y. The Periods of The Pell-p Orbits of Polyhedral and Centro-Polyhedral Groups. Jordan J. Math. Statistics. 10(1), 2017. 1-9.
• [7] Deveci O., Akuzum Y., Karaduman E., Erdag O. The Cyclic Groups via Bezout Matrices. J. Math. Research. 7(2), 2015, 34-41.
• [8] Deveci O., Shannon, A.G. The complex-type k-Fibonacci sequences and their applications. Commun. Algebra, 49(3), 2021, 1352- 1367.
• [9] Doostie H., Hashemi M. Fibonacci lengths involving the Wall number K(n). J. Appl. Math. Comput. 20(1), 2006, 171-180.
• [10] Erdag, O., Deveci O. The arrowhead-Pell-random-type sequences in finite groups. AIP Conference Proceedings. 1991(1), AIP Publishing LLC, 2018.
• [11] Falcon S., Plaza A. k-Fibonacci sequences modulo m. Chaos Solitons Fractals, 41(1), 2009, 497-504.
• [12] Gultekin, I., Deveci O. On the arrowhead-Fibonacci numbers. Open Math. 14(1), 2016, 1104-1113.
• [13] Karaduman E., Aydin H. k-nacci sequences in some special groups of finite order. Math. Comput. Modell. 50(1-2), 2009, 53-58.
• [14] Knox S.W. Fibonacci sequences in finite groups. Fibonacci Quart. 30(2), 1992, 116-120.
• [15] Lu K., Wang J. k-step Fibonacci sequence modulo m. Util. Math. 71, 2006, 169-177.
• [16] Ozkan E., Aydin H., Dikici, R. 3-step Fibonacci series modulo m. Appl. Math. Comput. 143(1), 2003, 165-172.
• [17] Robinson D.W. The Fibonacci matrix modulo m. Fibonacci Quart. 1, 1963, 29–36.
• [18] Shah A.P. Fibonacci sequence modulo m. Fibonacci Quart. 6(2), 1968, 139–141.
• [19] Tas S., Karaduman E. The Padovan sequences in finite groups. Chaing Mai J. Sci. 41(2), 2014, 456-462.
• [20] Wall D.D. Fibonacci series modulo m. American Math.Monthly. 67(6), 1960, 525-532.
• [21] Wilcox H.J. Fibonacci sequences of period n in Groups. Fibonacci Quart., 24(4), 1986, 356-361.