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A complete formula for the order of appearance of the powers of Lucas numbers

Prapanpong Pongsriiam
대한수학회 (The Korean Mathematical Society)|2016.07|대한수학회논문집 (Communications of the KMS)|제31권 제3호|pp. 447-450

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키워드

Fibonacci numberLucas numberthe order of appearancethe rank of appearance
주제분류수학

초록

Let $F_{n}$ and $L_n$ be the $n$th 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 $z(L_n^k)$ in some cases. In this article, we obtain the formula of $z(L_n^k)$ for all $n, k\geq 1$.

Abstract

Let $F_{n}$ and $L_n$ be the $n$th 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 $z(L_n^k)$ in some cases. In this article, we obtain the formula of $z(L_n^k)$ for all $n, k\geq 1$.

참고문헌 (8)

  1. B. A. Brousseau, "Fibonacci and Related Number Theoretic Tables" (1972), The Fibonacci Association
  2. J. H. Halton, "On the divisibility properties of Fibonacci numbers", Fibonacci Quart, 4(3), pp.217~240 (1966)
  3. T. Koshy, "Fibonacci and Lucas Numbers with Applications" (2001), Wiley
  4. T. Lengyel, "The order of the Fibonacci and Lucas Numbers", Fibonacci Quart, 33(3), pp.234~239 (1995)
  5. D. Marques, "The order of appearance of powers of Fibonacci and Lucas numbers", Fi-bonacci Quart, 50(3), pp.239~245 (2012)
  6. D. Marques, "The order of appearance of integers at most one away from Fibonacci numbers", Fibonacci Quart, 50(1), pp.36~43 (2012)
  7. P. Pongsriiam, "Exact divisibility by powers of the Fibonacci and Lucas numbers", J. Integer Seq, 17(11), pp.12~ (2014)
  8. S. Vajda, "Fibonacci and Lucas Numbers and the Golden Section: Theory and Applica-tions" (2007), Dover Publications

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