Coincidence of modified Pell numbers and sums of two Jacobsthal numbers
*Ahmed GaberCorresponding authora.gaber@sci.asu.edu.egAffiliation 1Department of MathematicsFaculty of ScienceAin Shams UniversityCairo, EgyptAffiliation 2Mathematics Education ProgramFaculty of Education and ArtsSohar UniversitySohar, 3111, Oman0000-0002-5620-901XView full profile → , Mohiedeen Ahmedmohyeeldinabdallah_p@sci.asu.edu.egDepartment of MathematicsFaculty of ScienceAin Shams UniversityCairo, EgyptView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 06 Sep 2023
- Published Online:
- 28 Feb 2025
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2138
- Pages:
- 205–216
Abstract
Keywords
Subject Classifications
References
[1] C. Adegbindin, F. Luca, and A. Togbé, “Pell and Pell-Lucas numbers as sums of two repdigits,” Bull. Malays. Math. Sci. Soc., vol. 43, pp. 1253-1271 (2020). [Online]. Available: https://doi.org/10.1007/s40840-019-00739-3.
[2] H. Bensella and D. Behloul, “Common terms of Leonardo and Jacobsthal numbers,” Rend. Circ. Mat. Palermo, II. Ser. vol. 43 (2023). [Online]. Available: https://doi.org/10.1007/s12215-023-00920-5.
[3] J. Bravo and F. Luca, “Perfect Pell and Pell-Lucas numbers,” Studia Sci. Math. Hungar., vol. 56, pp. 381-387 (2019). [Online]. Available: https://doi.org/10.1556/012.2019.56.4.1440.
[4] Y. Bugeaud, Linear forms in logarithms and applications, European Mathematical Society, Zurich (2018).
[5] Y. Bugeaud, M. Mignotte, and S. Siksek, “Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas powers,” Ann. Math., vol. 163, pp. 969-1018 (2006). [Online]. Available: https://doi.org/10.4007/annals.2006.163.969.
[6] A. Cağman, “An approach to Pillai’s problem with the Pell sequence and the powers of 3,” Miskolc Math. Notes, vol. 22, pp. 599-610 (2021). [Online]. Available: https://doi.org/10.18514/MMN.2021.3659.
[7] P. Catarino and H. Campos, “Incomplete k-Pell, k-Pell-Lucas and Modified k-Pell numbers,” Hacettepe Journal of Mathematics and Statistics, vol. 46, pp. 361-372 (2017). [Online]. Available: https://doi.org/10.15672/HJMS.20164518616.
[8] P. Catarino and H. Campos, “A note on Gaussian Modified Pell numbers,” Journal of Information and Optimization Sciences, vol. 39, pp. 1363-1371 (2018). [Online]. Available: https://doi.org/10.1080/02522667.2018.1471267.
[9] H. Cohen, Number theory, vol. 1, Springer, New York (2007).
[10] A. Dujella and A. Pethö, “A generalization of a theorem of Baker and Davenport,” Q. J. Math., vol. 49, pp. 291-306 (1998). [Online]. Available: https://doi.org/10.1093/qmathj/49.3.291.
[11] V. Facó and D. Marques, “Even perfect numbers among generalized Fibonacci sequences,” Rend. Circ. Mat. Palermo, II. Ser. vol. 63, pp. 363-370 (2014). [Online]. Available: https://doi.org/10.1007/s12215-014-0165-7.
[12] B. Faye and B. Edjeou, “On the problem of Pillai with Pell numbers, Pell-Lucas numbers and powers of 3,” Int. J. Number Theory, vol. 2, pp. 1-22 (2022). [Online]. Available: https://doi.org/10.1142/S1793042123500045.
[13] A. F. Horadam, “Jacobsthal representation numbers,” Fibonacci Quart., vol. 34, pp. 40-54 (1996).
[14] B. Kafle, S. E. Rihane, and A. Togbé, “Pell and Pell-Lucas numbers of the form xa ± xb + 1,” Bol. Soc. Mat. Mex., vol. 26, pp. 879-893 (2020). [Online]. Available: https://doi.org/10.1007/s40590-020-00305-.
[15] T. Koshy, Pell and Pell-Lucas numbers with applications, Springer, Berlin (2014).
[16] T. Koshy, Fibonacci and Lucas numbers with applications, vol. 2, 2nd ed., Wiley & Sons, Inc. (2019).
[17] E. M. Matveev, “An explicit lower bound for a homogeneous rational form in the logarithms of algebraic numbers. II,” Izv. Math., vol. 64, pp. 1217-1269 (2000). [Online]. Available: https://doi.org/10.1070/im2000v064n06abeh000314.
[18] Y. Y. Qu and J. W. Zeng, “Pell and Pell-Lucas numbers of the form −2a − 3b + 5c,” Czechoslovak Math. J., vol. 70, pp. 281-289 (2020). [Online]. Available: https://doi.org/10.21136/CMJ.2019.0265-18.
[19] B. Radičić, “On k-circulant matrices involving the Pell-Lucas (and the modified Pell) numbers,” Comp. Appl. Math., vol. 40, pp. 1-15 (2021), article 111. [Online]. Available: https://doi.org/10.1007/s40314-021-01473-y.
