Blocks within the period of Lucas sequence

Authors

  • Rima P. Patel Department of Social Science, Humanity and Mathematics, Mahavir Swami College of Polytechnic, Bhagwan Mahavir University, Surat-395017.
  • Dr. Devbhadra V. Shah Department of Mathematics, Veer Narmad South Gujarat University, Surat-395007.

DOI:

https://doi.org/10.23755/rm.v41i0.664

Keywords:

Fibonacci sequence, Lucas sequence, Periodicity of Lucas sequence

Abstract

In this paper, we consider the periodic nature of the sequence of  Lucas numbers L_n defined by the recurrence relation L_n= L_(n-1)+L_(n-2); for all n≥2; with initial condition L_0=2 and L_1=1. For any modulo m>1, we introduce the ‘blocks’ within this sequence by observing the distribution of residues within a single period of Lucas sequence. We show that length of any one period of the Lucas sequence contains either 1,2 or 4 blocks. 

References

Kramer J., Hoggatt. Jr. V. E., Special cases of Fibonacci Periodicity, Fibonacci Quarterly, 1:5, 1972, 519 – 522.

Marc R., Properties of the Fibonacci sequence under various moduli, Master’s Thesis, Wake Forest University, 1996.

Patel R. P., Shah D. V., Periodicity of generalized Lucas numbers and the length of its period under modulo 2^e, Mathematics Today, 33, 2017, 67 – 74.

Robinson D. W., The Fibonacci matrix modulo m, The Fibonacci Quarterly, 1, 29 – 36, 1963.

Thomas K., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Inc., New York, 2001.

Wall D. D., Fibonacci series modulo m, The American Mathematical Monthly, 67, 1960, 525 – 532.

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Published

2021-12-31

Issue

Vol. 41 (2021)

Section

Volume 41-2021

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