Divisibility of Fibonomial coefficients in terms of their digital representations and applications

Phakhinkon Napp Phunphayap; Prapanpong Pongsriiam; Phakhinkon Napp Phunphayap; Prapanpong Pongsriiam

doi:10.3934/math.2022296

AIMS Mathematics

2022, Volume 7, Issue 4: 5314-5327. doi: 10.3934/math.2022296

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Research article

Divisibility of Fibonomial coefficients in terms of their digital representations and applications

Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand
^aNapp is his nickname his parents gave him and he would like to use it as a middle name too. His first and last names read like Pa-kin-gorn Poon-pa-yap. He is the same person as Phakhinkon Phunphayap, one of the authors of the articles ^[13,14]

Received: 24 November 2021 Revised: 31 December 2021 Accepted: 03 January 2022 Published: 06 January 2022
MSC : 11B39, 11A63, 11N37, 11B65

We give a characterization for the integers $ n \geq 1 $ such that the Fibonomial coefficient $ {pn \choose n}_F $ is divisible by $ p $ for any prime $ p \neq 2, 5 $. Then we use it to calculate asymptotic formulas for the number of positive integers $ n \leq x $ such that $ p \mid {pn \choose n}_F $. This completes the study on this problem for all primes $ p $.
- Fibonomial coefficient,
- Fibonacci number,
- digit,
- sum of digits function,
- divisibility,
- asymptotic,
- binomial coefficient
Citation: Phakhinkon Napp Phunphayap, Prapanpong Pongsriiam. Divisibility of Fibonomial coefficients in terms of their digital representations and applications[J]. AIMS Mathematics, 2022, 7(4): 5314-5327. doi: 10.3934/math.2022296

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Abstract

We give a characterization for the integers $ n \geq 1 $ such that the Fibonomial coefficient $ {pn \choose n}_F $ is divisible by $ p $ for any prime $ p \neq 2, 5 $. Then we use it to calculate asymptotic formulas for the number of positive integers $ n \leq x $ such that $ p \mid {pn \choose n}_F $. This completes the study on this problem for all primes $ p $.

References

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