Let $ \left\{E_{n}\right\}_{n\geq0} $ and $ \left\{P_{n}\right\}_{n\geq0} $ be sequences of Perrin and Padovan numbers, respectively. We have found all repdigits that can be written as the sum or product of $ E_{n} $ and $ P_{m} $ in the base $ \eta $, where $ 2\leq\eta\leq10 $ and $ m\leq n $. In addition, we have applied the theory of linear forms in logarithms of algebraic numbers and Baker-Davenport reduction method in Diophantine approximation approaches.
Citation: Hunar Sherzad Taher, Saroj Kumar Dash. Repdigits base $ \eta $ as sum or product of Perrin and Padovan numbers[J]. AIMS Mathematics, 2024, 9(8): 20173-20192. doi: 10.3934/math.2024983
Let $ \left\{E_{n}\right\}_{n\geq0} $ and $ \left\{P_{n}\right\}_{n\geq0} $ be sequences of Perrin and Padovan numbers, respectively. We have found all repdigits that can be written as the sum or product of $ E_{n} $ and $ P_{m} $ in the base $ \eta $, where $ 2\leq\eta\leq10 $ and $ m\leq n $. In addition, we have applied the theory of linear forms in logarithms of algebraic numbers and Baker-Davenport reduction method in Diophantine approximation approaches.
| [1] | A. C. G. Lomelí, S. H. Hernández, Repdigits as sums of two Padovan numbers, J. Integer Seq., 22 (2019), Article 19.2.3. |
| [2] |
P. Trojovský, On repdigits as sums of Fibonacci and Tribonacci numbers, Symmetry, 12 (2020), 1774. https://doi.org/10.3390/sym12111774 doi: 10.3390/sym12111774
|
| [3] |
D. Bednařík, E. Trojovská, Repdigits as product of Fibonacci and Tribonacci numbers, Mathematics, 8 (2020), 1720. https://doi.org/10.3390/math8101720 doi: 10.3390/math8101720
|
| [4] |
F. Erduvan, R. Keskin, Z. Şiar, Repdigits base $b$ as products of two Pell numbers or Pell–Lucas numbers, Bol. Soc. Mat. Mex., 27 (2021), 70. https://doi.org/10.1007/s40590-021-00377-5 doi: 10.1007/s40590-021-00377-5
|
| [5] |
F. Erduvan, R. Keskin, Z. Şiar, Repdigits base $b$ as products of two Lucas numbers, Quaestiones Mathematicae, 44 (2021), 1283–1293. https://doi.org/10.2989/16073606.2020.1787539 doi: 10.2989/16073606.2020.1787539
|
| [6] |
F. Erduvan, R. Keskin, Z. Şiar, Repdigits base $b$ as products of two Fibonacci numbers, Indian J. Pure Appl. Math., 52 (2021), 861–868. https://doi.org/10.1007/s13226-021-00041-8 doi: 10.1007/s13226-021-00041-8
|
| [7] |
Z. Şiar, F. Erduvan, R. Keskin, Repdigits base $b$ as difference of two Fibonacci numbers, J. Math. Study, 55 (2022), 84–94. https://doi.org/10.4208/jms.v55n1.22.07 doi: 10.4208/jms.v55n1.22.07
|
| [8] |
K. Bhoi, P. K. Ray, Perrin numbers expressible as sums of two base $b$ repdigits, Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics, 53 (2021), 1–11. https://doi.org/10.13137/2464-8728/33226 doi: 10.13137/2464-8728/33226
|
| [9] |
S. E. Rihane, A. Togbé, Repdigits as products of consecutive Padovan or Perrin numbers, Arab. J. Math., 10 (2021), 469–480. https://doi.org/10.1007/s40065-021-00317-1 doi: 10.1007/s40065-021-00317-1
|
| [10] |
S. E. Rihane, A. Togbé, Padovan and Perrin numbers as product of two repdigits, Bol. Soc. Mat. Mex., 28 (2022), 51. https://doi.org/10.1007/s40590-022-00446-3 doi: 10.1007/s40590-022-00446-3
|
| [11] |
K. N. Adédji, A. Filipin, A. Togbé, Padovan and Perrin numbers which are products of two repdigits in base $b$, Results Math., 76 (2021), 193. https://doi.org/10.1007/s00025-021-01502-6 doi: 10.1007/s00025-021-01502-6
|
| [12] |
K. N. Adédji, V. Dossou-yovo, S. E. Rihane, A. Togbé, Padovan or Perrin numbers that are concatenations of two distinct base $b$ repdigits, Math. Slovaca, 73 (2023), 49–64. https://doi.org/10.1515/ms-2023-0006 doi: 10.1515/ms-2023-0006
|
| [13] |
K. N. Adédji, J. Odjoumani, A. Togbé, Padovan and Perrin numbers as products of two generalized Lucas numbers, Arch. Math., 59 (2023), 315–337. https://doi.org/10.5817/AM2023-4-315 doi: 10.5817/AM2023-4-315
|
| [14] |
M. G. Duman, R. Keskin, L. Hocaoğlu, Padovan numbers as sum of two repdigits, C. R. Acad. Bulg. Sci., 76 (2023), 1326–1334. https://doi.org/10.7546/CRABS.2023.09.02 doi: 10.7546/CRABS.2023.09.02
|
| [15] |
E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. Ⅱ, Izvestiya: Mathematics, 64 (2000), 1217–1269. https://doi.org/10.1070/im2000v064n06abeh000314 doi: 10.1070/im2000v064n06abeh000314
|
| [16] |
Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations Ⅰ. Fibonacci and Lucas perfect powers, Ann. Math., 163 (2006), 969–1018. https://doi.org/10.4007/annals.2006.163.969 doi: 10.4007/annals.2006.163.969
|
| [17] | B. M. M. De Weger, Algorithms for Diophantine equations, Amsterdam, Netherlands: Centrum voor Wiskunde en Informatica, 1989. |
| [18] |
S. G. Sanchez, F. Luca, Linear combinations of factorials and $s$-units in a binary recurrence sequence, Ann. Math. Québec, 38 (2014), 169–188. https://doi.org/10.1007/s40316-014-0025-z doi: 10.1007/s40316-014-0025-z
|
Hunar Sherzad Taher, Saroj Kumar Dash. Repdigits base $ \eta $ as sum or product of Perrin and Padovan numbers[J]. AIMS Mathematics, 2024, 9(8): 20173-20192. doi: 10.3934/math.2024983
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