The First Study of Mersenne--Leonardo Sequence

Eudes Antonio Costa; Paula Maria Machado Cruz Catarino

doi:10.33434/cams.1598817

EN

The First Study of Mersenne--Leonardo Sequence

Abstract

In this study, we introduce a new class of numbers, referred to as Modified Mersenne--Leonardo numbers. The aim of this paper is to define the Modified Mersenne--Leonardo sequence and investigate some of its properties, including the recurrence relation, summation formula, and generating function. Additionally, classical identities such as the Tagiuri–Vajda, Catalan, Cassini, and d’Ocagne identities are derived for the Modified Mersenne--Leonardo numbers.

Keywords

Binet’s formula , Generating function , Leonardo numbers , Mersenne numbers , Mersenne--Leonardo numbers

References

[1] J. A. N. Sloane and others. The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation Inc., (2024). https://oeis.org/
[2] E. Deza. Mersenne numbers and Fermat numbers. World Scientific, 1 (2021).
[3] J. S. Hall, A Remark on the Primeness of Mersenne Numbers, J. London Math. Soc., 1(3) (1953), 285–287.
[4] B.A. Delello, Perfect Numbers and Mersenne primes, PhD thesis, University of Central Florida, 1986.
[5] A. J. Gomes, E. A. Costa, R. A. Santos, Numeros perfeitos e primos de Mersenne, Revista da Olimpiada, IME-UFG, 7 (2008), 99–111.
[6] D. Aggarwal, A. Joux, A. Prakash, Anupam, M. Santha, A new public-key cryptosystem via Mersenne numbers, Advances in Cryptology–CRYPTO 2018: 38th Annual International Cryptology Conference, (2018), 459–482.
[7] C. J. L. Padmaja, V.S. Bhagavan, B. Srinivas, RSA encryption using three Mersenne primes, Int. J. Chem. Sci, 14(4) (2016), 2273–2278.
[8] P. Catarino, H. Campos, Helena, P. Vasco, On the Mersenne sequence, Ann. Math. Inform., 46 (2016), 37–53.
[9] M. Chelgham, A. Boussayoud, On the k-Mersenne-Lucas numbers, Notes Number Theory Discrete Math., 27(1) (2021), 7–13.
[10] M. Mangueira, R. Vieira, F. Alves, P. Catarino, As generalizacoes das formas matriciais e dos quaternionos da sequencia de Mersenne, Rev. Mat. Univ. Fed. Ouro Preto., 1(1) (2021), 1–17.

[11] Y. Soykan, A study on generalized Mersenne numbers, J. Progressive Research Math., 18(3) (2021), 90–112.
[12] C. King, Charles, Leonardo Fibonacci, Fibonacci Quart., 1(4) (1963), 15–20.
[13] E. W. Dijkstra, Fibonacci Numbers and Leonardo Numbers, EWD-797, University of Texas at Austin, (1981).
[14] E. W. Dijkstra, Smoothsort, an alternative for sorting in situ, Sci. Comput. Programming, 1(3) (1982), 223–233.
[15] P. Catarino, A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian., 89(1) (2019), 75–86.
[16] F. R. Alves, R .P. Vieira, The Newton fractal’s Leonardo sequence study with the Google Colab, Int. Electron. J. Math. Edu., 15(2) (2019), em0575.
[17] Y. Alp, E. G. Koçer, Some properties of Leonardo numbers, Konuralp J. Math., 9(1) (2021), 183–189. (Dergipark).
[18] P. Beites, P. Franc¸a, C. Moreira, Uma f´ormula de tipo Binet para os numeros de Geonardo, Gazeta de Matematica, 201 (2023), 20–23.
[19] P. Beites, P. Catarino, On the Leonardo Quaternions Sequence, Hacet. J. Math. Stat., 3 (4) (2023), 1001–1023.
[20] N. Kara, F. Yilmaz, On hybrid numbers with Gaussian Leonardo coefficients, Mathematics, 11(6) (2023), 1551.
[21] A. G. Shannon, A note on generalized Leonardo numbers, Notes Number Theory Discrete Math., 25(3) (2019), 97–101.
[22] E. Tan, H. Leung, On Leonardo p-numbers, Integers, 23 (2023), 1-11.
[23] E.V. Spreafico, E.A. Costa, P. Catarino, Hybrid Numbers with Hybrid Leonardo Number Coefficients. J. Math. Ext., 18 (2) (2024), 1-17.
[24] Ç. Çelemoğlu, Pell Leonardo numbers and their matrix representations, J. New Results Sci., 13(2) (2024), 101–108.
[25] E. Özkan, H. Akkuş, Generalized Bronze Leonardo sequence, Notes Number Theory Discrete Math., 30(4) (2024), 811–824.
[26] A. Özkoç Öztürk, V. Külahlı, Cobalancing numbers: Another way of demonstrating their properties, Commun. Adv. Math. Sci., 7(1) (2024), 1–13.
[27] A. F. Horadam. A generalized Fibonacci sequence, The American Mathematical Monthly, 68(5) (1961), 455–459.
[28] D. Kalman, R. Mena, The Fibonacci numbers—exposed, Math. Mag., 76(3) (2003), 167–181.

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Eudes Antonio Costa ^*
0000-0001-6684-9961
Brazil

Paula Maria Machado Cruz Catarino
0000-0001-6917-5093
Portugal

Early Pub Date

February 25, 2025

Publication Date

March 27, 2025

Submission Date

December 9, 2024

Acceptance Date

January 27, 2025

Published in Issue

Year 2025 Volume: 8 Number: 1

DOI

https://doi.org/10.33434/cams.1598817

IZ

https://izlik.org/JA39WA97LL

APA

Costa, E. A., & Catarino, P. M. M. C. (2025). The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences, 8(1), 11-23. https://doi.org/10.33434/cams.1598817

AMA

1.Costa EA, Catarino PMMC. The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences. 2025;8(1):11-23. doi:10.33434/cams.1598817

Chicago

Costa, Eudes Antonio, and Paula Maria Machado Cruz Catarino. 2025. “The First Study of Mersenne--Leonardo Sequence”. Communications in Advanced Mathematical Sciences 8 (1): 11-23. https://doi.org/10.33434/cams.1598817.

EndNote

Costa EA, Catarino PMMC (March 1, 2025) The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences 8 1 11–23.

IEEE

[1]E. A. Costa and P. M. M. C. Catarino, “The First Study of Mersenne--Leonardo Sequence”, Communications in Advanced Mathematical Sciences, vol. 8, no. 1, pp. 11–23, Mar. 2025, doi: 10.33434/cams.1598817.

ISNAD

Costa, Eudes Antonio - Catarino, Paula Maria Machado Cruz. “The First Study of Mersenne--Leonardo Sequence”. Communications in Advanced Mathematical Sciences 8/1 (March 1, 2025): 11-23. https://doi.org/10.33434/cams.1598817.

JAMA

1.Costa EA, Catarino PMMC. The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences. 2025;8:11–23.

MLA

Costa, Eudes Antonio, and Paula Maria Machado Cruz Catarino. “The First Study of Mersenne--Leonardo Sequence”. Communications in Advanced Mathematical Sciences, vol. 8, no. 1, Mar. 2025, pp. 11-23, doi:10.33434/cams.1598817.

Vancouver

1.Eudes Antonio Costa, Paula Maria Machado Cruz Catarino. The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences. 2025 Mar. 1;8(1):11-23. doi:10.33434/cams.1598817

Cited By

Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms

Mathematical Sciences and Applications E-Notes

https://doi.org/10.36753/mathenot.1797254

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The First Study of Mersenne--Leonardo Sequence

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References

Details

Primary Language

Subjects

Journal Section

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Early Pub Date

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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Cited By

Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms