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Yıl 2025, , 11 - 23, 27.03.2025

Eudes Antonio Costa , Paula Maria Machado Cruz Catarino

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

Öz

Kaynakça

  • [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.

The First Study of Mersenne--Leonardo Sequence

Yıl 2025, , 11 - 23, 27.03.2025

Eudes Antonio Costa , Paula Maria Machado Cruz Catarino

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • [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.
Toplam 28 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Temel Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Eudes Antonio Costa 0000-0001-6684-9961

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

Erken Görünüm Tarihi 25 Şubat 2025
Yayımlanma Tarihi 27 Mart 2025
Gönderilme Tarihi 9 Aralık 2024
Kabul Tarihi 27 Ocak 2025
Yayımlandığı Sayı Yıl 2025

Kaynak Göster

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 Costa EA, Catarino PMMC. The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences. Mart 2025;8(1):11-23. doi:10.33434/cams.1598817
Chicago Costa, Eudes Antonio, ve Paula Maria Machado Cruz Catarino. “The First Study of Mersenne--Leonardo Sequence”. Communications in Advanced Mathematical Sciences 8, sy. 1 (Mart 2025): 11-23. https://doi.org/10.33434/cams.1598817.
EndNote Costa EA, Catarino PMMC (01 Mart 2025) The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences 8 1 11–23.
IEEE E. A. Costa ve P. M. M. C. Catarino, “The First Study of Mersenne--Leonardo Sequence”, Communications in Advanced Mathematical Sciences, c. 8, sy. 1, ss. 11–23, 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 (Mart 2025), 11-23. https://doi.org/10.33434/cams.1598817.
JAMA Costa EA, Catarino PMMC. The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences. 2025;8:11–23.
MLA Costa, Eudes Antonio ve Paula Maria Machado Cruz Catarino. “The First Study of Mersenne--Leonardo Sequence”. Communications in Advanced Mathematical Sciences, c. 8, sy. 1, 2025, ss. 11-23, doi:10.33434/cams.1598817.
Vancouver Costa EA, Catarino PMMC. The First Study of Mersenne--Leonardo Sequence. Communications in Advanced Mathematical Sciences. 2025;8(1):11-23.

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