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The order of appearance of the product of two Fibonacci and Lucas numbers

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Abstract

Let \(F_{n}\) and \(L_{n}\) be the nth Fibonacci and Lucas number, respectively. The order of appearance is defined as the smallest natural number k such that n divides \(F_{k}\) and denoted by z(n) . In this paper, we give explicit formulas for the terms \( z(F_{a}F_{b}) \), \( z( L_{a}L_{b}) \), \( z(F_{a}L_{b}) \) and \( z(F_{n}F_{n+p}F_{n+2p}) \) with \(p\ge 3\) prime.

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Acknowledgement

The authors are grateful to the reviewer for helpful and valuable comments and remarks. Thanks to the comments and remarks, we added the part Theorem 1.1(iii) with its proof.

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Authors and Affiliations

  1. Institute of Mathematics, Niğde Ömer Halisdemir University, Niğde, Turkey

    N. Irmak

  2. Sambalpur University, Jyoti Vihar, Burla, India

    P. k. Ray

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  1. N. Irmak
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  2. P. k. Ray
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Correspondence to N. Irmak.

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Irmak, N., Ray, P.k. The order of appearance of the product of two Fibonacci and Lucas numbers. Acta Math. Hungar. 162, 527–538 (2020). https://doi.org/10.1007/s10474-020-01052-3

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  • DOI: https://doi.org/10.1007/s10474-020-01052-3

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