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THE PRODUCT OF THREE k-GENERALIZED LUCAS NUMBERS AS REPDIGITS
Keywords:
k-generalized Lucas sequences, k-Lucas numbers, linear forms in logarithms, Lucas numbersDOI:
https://doi.org/10.17654/0972555525032Abstract
In this paper, we search all repdigits which are products of arbitrary three terms of $k$-generalized Lucas sequences. Thus, we find all nonnegative integer solutions of the Diophantine equation $L_n^{(k)} L_m^{(k)} L_l^{(k)}=a\left(\frac{10^d-1}{9}\right)$, where $n \geq m \geq l \geq 0$ and $1 \leq a \leq 9$ for all $k \geq 2$ and $d \geq 1$.
Received: June 18, 2025
Revised: July 3, 2025
Accepted: July 18, 2025
References
[1] A. Alahmadi, A. Altassan, F. Luca and H. Shoaib, Products of k-Fibonacci numbers which are rep-digits, Publicationes Mathematicae 97 (2020), 1-2.
[2] M. Alan and K. S. Alan, Products of three Fibonacci numbers that are repdigits, Advanced Studies: Euro-Tbilisi Mathematical Journal 16 (2023), 57-66.
[3] A. Alahmadi, A. Altassan, F. Luca and H. Shoaib, k-generalized Fibonacci numbers which are concatenations of two repdigits, Glasnik Matematički 56 (2021), 29-46.
[4] A. Altassan and M. Alan, Mersenne numbers in generalized Lucas sequences, Proceedings of the Bulgarian Academy of Sciences 77 (2024), 3-10.
[5] D. Bednařík and E. Trojovská, Repdigits as product of Fibonacci and Tribonacci numbers, Mathematics 8(10) (2020), 1720.
[6] E. F. Bravo, J. J. Bravo and C. A. Gómez, Generalized Lucas numbers which are concatenations of two repdigits, Results in Mathematics 76 (2021), 1-16.
[7] J. J. Bravo, C. A. Gómez and F. Luca, Arithmetic properties of generalized Fibonacci numbers, Revista de la Unión Matemática Argentina 64 (2023), 439-460.
[8] J. J. Bravo, C. A. G. Gómez and F. Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Math. Notes 17 (2016), 85-100.
[9] J. J. Bravo and F. Luca, Repdigits in k-Lucas sequences, Proc. Indian Acad. Sci. Math. Sci. 124 (2014), 141-154.
[10] A. Çağman, Repdigits as product of Fibonacci and Pell numbers, Turkish Journal of Science 6 (2021), 31-35.
[11] P. Coufal and P. Trojovsky, Repdigits as product of terms of k-Bonacci sequences, Mathematics 9(6) (2021), 682.
[12] A. Dujella and A. Pethö, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. 49 (1998), 291-306.
[13] F. Erduvan and R. Keskin, Repdigits as products of two Fibonacci or Lucas numbers, Proc. Indian Acad. Sci. (Math. Sci.) 130 (2020), Article 28.
[14] F. Erduvan, R. Keskin and Z. Şiar, Repdigits base b as products of two Fibonacci numbers, Indian Journal of Pure and Applied Mathematics 52 (2021), 861-868.
[15] F. Erduvan, R. Keskin and Z. Şiar, Repdigits base b as products of two Lucas numbers, Quaestiones Mathematicae 44(10) (2021), 1283-1293.
[16] F. Erduvan, R. Keskin and Z. Şiar, Repdigits base b as products of two Pell numbers or Pell-Lucas numbers, Boletín de la Sociedad Matemática Mexicana 27 (2021), Article No. 70.
[17] S. Guzman and F. Luca, Linear combinations of factorials and S-units in a Binary Recurrence Sequence, Ann. Math. Québec 38 (2014), 169-188.
[18] N. Irmak and A. Togbe, On repdigits as product of consecutive Lucas numbers, Notes on Number Theory and Discrete Mathematics 24 (2018), 95-102.
[19] F. Luca, Fibonacci and Lucas numbers with only one distinct digit, Portugaliae Mathematica 57 (2000), 243-254.
[20] D. Marques, On k-generalized Fibonacci numbers with only one distinct digit, Util. Math. 98 (2015), 23-31.
[21] 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. Translation in Izv. Math. 64 (2000), 1217-1269.
[22] S. Seffah, S. E. Rihane, Salah Eddine and A. Togbe, On repdigits as product of k Fibonacci and k-Lucas numbers, Recall 1 (2025), 1.
[23] Z. Şiar, F. Erduvan and R. Keskin, Repdigits as products of two Pell or Pell-Lucas numbers, Acta Mathematica Universitatis Comenianae 88 (2019), 247-256.
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