On Padovan numbers that are perfect powers of Lucas numbers

Fatih Erduvan; Merve Güney Duman; Refik Keskin; Fatih Erduvan; Merve Güney Duman; Refik Keskin

doi:10.3934/math.2025733

AIMS Mathematics

2025, Volume 10, Issue 7: 16393-16406. doi: 10.3934/math.2025733

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Research article

On Padovan numbers that are perfect powers of Lucas numbers

1.
MEB, İzmit Namik Kemal Anatolian High School, Kocaeli, Türkiye
2.
Sakarya University of Applied Sciences, Department of Engineering Fundamental Science, Sakarya, Türkiye
3.
Sakarya University, Department of Mathematics, Sakarya, Türkiye

Received: 13 February 2025 Revised: 24 May 2025 Accepted: 30 June 2025 Published: 21 July 2025
MSC : 11B83, 11D61, 11J86

Let $ (P_{n}) $ and $ (L_{n}) $ be the Padovan and Lucas sequences. In this paper, we found all Padovan numbers that were perfect powers of Lucas numbers using advanced mathematical tools such as linear forms in logarithms and the reduction method. The results for this equation were
$ \begin{equation*} 9 = P_{9} = L_{2}^{2} = 3^{2}, 16 = P_{11} = L_{3}^{2} = 4^{2}, 49 = P_{15} = L_{4}^{2} = 7^{2}. \end{equation*} $
Moreover, the intersection between the Padovan and Lucas sequences revealed that
$ \begin{equation*} 1 = P_{2} = L_{1}, 2 = P_{3} = L_{0}, 2 = P_{4} = L_{0}, 3 = P_{5} = L_{2}, 4 = P_{6} = L_{3}, 7 = P_{8} = L_{4}. \end{equation*} $
This work not only solved a specific open problem in exponential Diophantine equations but also highlighted the wider applicability of the methods used. The results paved the way for further research on perfect powers in other linear iteration sequences and their intersections, emphasizing the importance of such research in modern mathematics.
- Padovan numbers,
- Lucas numbers,
- exponential Diophantine equations,
- Baker's method
Citation: Fatih Erduvan, Merve Güney Duman, Refik Keskin. On Padovan numbers that are perfect powers of Lucas numbers[J]. AIMS Mathematics, 2025, 10(7): 16393-16406. doi: 10.3934/math.2025733

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Abstract

Let $ (P_{n}) $ and $ (L_{n}) $ be the Padovan and Lucas sequences. In this paper, we found all Padovan numbers that were perfect powers of Lucas numbers using advanced mathematical tools such as linear forms in logarithms and the reduction method. The results for this equation were

$ \begin{equation*} 9 = P_{9} = L_{2}^{2} = 3^{2}, 16 = P_{11} = L_{3}^{2} = 4^{2}, 49 = P_{15} = L_{4}^{2} = 7^{2}. \end{equation*} $

Moreover, the intersection between the Padovan and Lucas sequences revealed that

$ \begin{equation*} 1 = P_{2} = L_{1}, 2 = P_{3} = L_{0}, 2 = P_{4} = L_{0}, 3 = P_{5} = L_{2}, 4 = P_{6} = L_{3}, 7 = P_{8} = L_{4}. \end{equation*} $

This work not only solved a specific open problem in exponential Diophantine equations but also highlighted the wider applicability of the methods used. The results paved the way for further research on perfect powers in other linear iteration sequences and their intersections, emphasizing the importance of such research in modern mathematics.

References

[1]	L. Moser, L. Carlitz, Advanced problem H-2, Fibonacci Quart., 1 (1963), 46. https://doi.org/10.1080/00150517.1963.12431600 doi: 10.1080/00150517.1963.12431600
[2]	A. P. Rollet, Advanced problem 5080, Am. Math. Mon., 70 (1963), 216. https://doi.org/10.2307/2312909 doi: 10.2307/2312909
[3]	J. H. E. Cohn, On square Fibonacci numbers, J. London Math. Soc., 39 (1964), 537–540. https://doi.org/10.1080/13594866409441744 doi: 10.1080/13594866409441744
[4]	O. Wyler, In the Fibonacci series $F_{1} = 1, F_{2} = 1, F_{n+1} = F_{n}+F_{n-1}$ the first, second and twelfth terms are squares, Amer. Math. Monthly, 71 (1964), 221–222. https://doi.org/10.2307/2311778 doi: 10.2307/2311778
[5]	B. U. Alfred, On square Lucas numbers, Fibonacci Quart., 2 (1964), 11–12.
[6]	H. London, R. Finkelstein, On Fibonacci and Lucas numbers which are perfect powers, Fibonacci Quart., 7 (1969), 476–481. https://doi.org/10.1080/00150517.1969.12431130 doi: 10.1080/00150517.1969.12431130
[7]	A. Pethő, Full cubes in the Fibonacci sequence, Publ. Math. Debrecen, 30 (1983), 117–127.
[8]	Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations Ⅰ. Fibonacci and Lucas perfect powers, Ann. Math., 163 (2016), 969–1018.
[9]	S. Karakaya, H. Özdemir, T. Petik, $3\times3$ Dimensional special matrices associated with Fibonacci and Lucas numbers, Sakarya Univ. J. Sci., 22 (2018), 1917–1922. https://doi.org/10.16984/saufenbilder.479060 doi: 10.16984/saufenbilder.479060
[10]	A. Kargın, E. Kişi, H. Özdemir, Some notes on odd or even indexed Fibonacci and Lucas sequences, Sakarya Univ. J. Sci., 23 (2019), 929–933. https://doi.org/10.1007/s12045-018-0695-y doi: 10.1007/s12045-018-0695-y
[11]	İ. Erduran, Z. Şiar, All solutions of the Diophantine equations $F_{n} = 3^{s}\cdot y^{b}$ and $F_{n}\pm1 = 3^{s}\cdot y^{b}$, Sakarya Univ. J. Sci., 26 (2022), 488–492. https://doi.org/10.16984/saufenbilder.1069960 doi: 10.16984/saufenbilder.1069960
[12]	Y. Taşyurdu, N. Ş. Türkoğlu, bi-periodic $(p, q)$ -Fibonacci and Bi-periodic $(p, q)$-Lucas sequences, Sakarya Univ. J. Sci., 27, (2023), 1–13. https://doi.org/10.16984/saufenbilder.1148618 doi: 10.16984/saufenbilder.1148618
[13]	A. Z. Azak, Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions, Mathematics, 10 (2022), 4655. https://doi.org/10.3390/math10244655 doi: 10.3390/math10244655
[14]	T. Erişir, M. A. Güngör, On Fibonacci spinors, Int. J. Geom. Methods M., 17 (2020), 2050065.
[15]	F. Erduvan, M. G. Duman, Mulatu numbers which are concatenation of two Fibonacci numbers, Sakarya Univ. J. Sci., 27 (2023), 1122–1127. https://doi.org/10.16984/saufenbilder.1235571 doi: 10.16984/saufenbilder.1235571
[16]	F. Erduvan, R. Keskin, Pell and Pell–Lucas numbers as product of two repdigits, Math. Notes, 112 (2022), 861–871. https://doi.org/10.1134/S0001434622110207 doi: 10.1134/S0001434622110207
[17]	Z. Şiar, R. Keskin, F. Erduvan, Fibonacci or Lucas numbers which are products of two repdigits in base $b$, B. Braz. Math. Soc., 52 (2021), 1025–1040. https://doi.org/10.1007/s00574-021-00243-y doi: 10.1007/s00574-021-00243-y
[18]	Y. Bugeaud, H. Kaneko, On perfect power in linear recurrence sequences of integers, Kyushu J. Math., 73 (2019), 221–227. https://doi.org/10.2206/kyushujm.73.221 doi: 10.2206/kyushujm.73.221
[19]	A. Pethő, Diophantine properties of linear recursive sequences Ⅱ, Acta Math. Acad. Paedagog. Nyhazi, 17 (2001), 81–96.
[20]	F. Erduvan, F. Luca, M. G. Duman, Z. F. Shadow, k-Generalized Fibonacci numbers which are perfect powers of Lucas numbers, Publ. Math.-Debrecen, in press.
[21]	H. Özdemir, S. Karakaya, On some properties of k-generalized Fibonacci numbers, Math. Commun., 29 (2024), 193–202.
[22]	Y. Bugeaud, Linear forms in logarithms and applications, IRMA Lectures in Mathematics and Theoretical Physics, Zurich: European Mathematical Society, 2018, 28.
[23]	E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers Ⅱ, Izv. Ross. Akad. Nauk Ser. Mat., 64, (2000), 125–180. (Russian) https://doi.org/10.4213/im314 doi: 10.4213/im314
[24]	J. J. Bravo, C. A. Gomez, F. Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Math. Notes, 17 (2016), 85–100.
[25]	A. Dujella, A. Pethò, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxf. Ser., 49 (1998), 291–306.
[26]	B. M. M. De Weger, Algorithms for Diophantine equations, CWI Tracts 65, Stichting Mathematisch Centrum, Amsterdam, 1989.
[27]	M. Ismail, A. Gaber, M. Anwar, Padovan and Perrin numbers as sums of two Jacobsthal numbers, arXiv: 2208.00276v1, 2022.

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