대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제31권3호
  • /
  • Pages.447-450
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  • 2016
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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A COMPLETE FORMULA FOR THE ORDER OF APPEARANCE OF THE POWERS OF LUCAS NUMBERS

  • 투고 : 2015.09.06
  • 발행 : 2016.07.31
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초록

Let $F_n$ and $L_n$ be the nth Fibonacci number and Lucas number, respectively. The order of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that m divides $F_k$. Marques obtained the formula of $z(L^k_n)$ in some cases. In this article, we obtain the formula of $z(L^k_n)$ for all $n,k{\geq}1$.

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참고문헌

  1. B. A. Brousseau, Fibonacci and Related Number Theoretic Tables, Santa Clara, The Fibonacci Association, 1972.
  2. J. H. Halton, On the divisibility properties of Fibonacci numbers, Fibonacci Quart. 4 (1966), no. 3, 217-240.
  3. T. Koshy, Fibonacci and Lucas Numbers with Applications, New York, Wiley, 2001.
  4. T. Lengyel, The order of the Fibonacci and Lucas Numbers, Fibonacci Quart. 33 (1995), no. 3, 234-239.
  5. D. Marques, The order of appearance of powers of Fibonacci and Lucas numbers, Fi-bonacci Quart. 50 (2012), no. 3, 239-245.
  6. D. Marques, The order of appearance of integers at most one away from Fibonacci numbers, Fibonacci Quart. 50 (2012), no. 1, 36-43.
  7. P. Pongsriiam, Exact divisibility by powers of the Fibonacci and Lucas numbers, J. Integer Seq. 17 (2014), no. 11, Article 14.11.2, 12 pp.
  8. S. Vajda, Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications, New York, Dover Publications, 2007.

피인용 문헌

  1. The order of appearance of factorials in the Fibonacci sequence and certain Diophantine equations pp.1588-2829, 2018, https://doi.org/10.1007/s10998-018-0268-6