Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON THE EXTENT OF THE DIVISIBILITY OF FIBONOMIAL COEFFICIENTS BY A PRIME NUMBER

  • Lee, David Taehee (Institude of Science Education for the Gifted and Talented, Yonsei University) ;
  • Lee, Juhyep (Institude of Science Education for the Gifted and Talented, Yonsei University) ;
  • Park, Jinseo (Department of Mathematics Education, Catholic Kwandong University)
  • Received : 2021.02.04
  • Accepted : 2021.12.01
  • Published : 2021.12.30
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Abstract

Let (Fn)n≥0 be the Fibonacci sequence and p be a prime number. For 1≤k≤m, the Fibonomial coefficient is defined as $$\[\array{m\\k}\]_F=\frac{F_{m-k+1}{\ldots}{F_{m-1}F_m}}{{F_1}{\ldots}{F_k}}$$ and $\[\array{m\\k}\]_F=0$ whan k>m. Let a and n be positive integers. In this paper, we find the conditions of prime number p which divides Fibonomial coefficient $\[\array{P^{a+n}\\{p^a}}\]_F$. Furthermore, we also find the conditions of p when $\[\array{P^{a+n}\\{p^a}}\]_F$ is not divisible by p.

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Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2019R1G1A1006396).

References

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  3. D. Margues, J. Sellers, and P. Trojovske, On divisibility properties of certain Fibonomial coefficients by a prime p, Fibonacci Quart. 51 (2013), 78-83