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On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes

Author

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  • Pavel Trojovský

    (Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic)

Abstract

The k -generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k -bonacci or k -step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the Diophantine equation F m ( k ) = m t , with t > 1 and m > k + 1 , has only solutions F 12 ( 2 ) = 12 2 and F 9 ( 3 ) = 9 2 .

Suggested Citation

  • Pavel Trojovský, 2019. "On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes," Mathematics, MDPI, vol. 7(8), pages 1-10, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:700-:d:254586
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    Citations

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    Cited by:

    1. Yunyun Qu & Jiwen Zeng, 2020. "Lucas Numbers Which Are Concatenations of Two Repdigits," Mathematics, MDPI, vol. 8(8), pages 1-8, August.
    2. Pavel Trojovský, 2020. "On the Characteristic Polynomial of the Generalized k -Distance Tribonacci Sequences," Mathematics, MDPI, vol. 8(8), pages 1-8, August.
    3. Pavel Trojovský & Štěpán Hubálovský, 2020. "Some Diophantine Problems Related to k -Fibonacci Numbers," Mathematics, MDPI, vol. 8(7), pages 1-10, June.
    4. Petr Coufal & Pavel Trojovský, 2021. "Repdigits as Product of Terms of k -Bonacci Sequences," Mathematics, MDPI, vol. 9(6), pages 1-10, March.
    5. Marie Hubálovská & Štěpán Hubálovský & Eva Trojovská, 2020. "On Homogeneous Combinations of Linear Recurrence Sequences," Mathematics, MDPI, vol. 8(12), pages 1-7, December.
    6. Dongwei Guo & Wenchang Chu, 2022. "Sums of Pell/Lucas Polynomials and Fibonacci/Lucas Numbers," Mathematics, MDPI, vol. 10(15), pages 1-10, July.
    7. Pavel Trojovský, 2020. "Fibonacci Numbers with a Prescribed Block of Digits," Mathematics, MDPI, vol. 8(4), pages 1-7, April.
    8. Eva Trojovská & Pavel Trojovský, 2021. "On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers," Mathematics, MDPI, vol. 9(9), pages 1-9, April.
    9. Muflih Alhazmi & Rexma Sherine Venchislas & Gerly Thaniel Gnanamuthu & Chellamani Perumal & Shreefa O. Hilali & Mashaer Alsaeedi & Avinash Natarajan & Britto Antony Xavier Gnanaprakasam, 2024. "H -Nacci Sequence and Its Role in Virus Mutation," Mathematics, MDPI, vol. 12(17), pages 1-19, August.
    10. Ana Paula Chaves & Pavel Trojovský, 2020. "A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers," Mathematics, MDPI, vol. 8(6), pages 1-10, June.
    11. Dušan Bednařík & Eva Trojovská, 2020. "Repdigits as Product of Fibonacci and Tribonacci Numbers," Mathematics, MDPI, vol. 8(10), pages 1-8, October.
    12. Nazlıhan Terzioğlu & Can Kızılateş & Wei-Shih Du, 2022. "New Properties and Identities for Fibonacci Finite Operator Quaternions," Mathematics, MDPI, vol. 10(10), pages 1-13, May.

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