Research Article
BibTex RIS Cite

Some Properties between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers

Year 2025, Volume: 13 Issue: 1, 21 - 27, 30.04.2025

Gamaliel Morales

Abstract

We define the Gauss Lichtenberg numbers. Then we give a formula for the Gauss Lichtenberg numbers by using the Lichtenberg numbers. We show that there is a relation between the Gauss Lichtenberg numbers, Lichtenberg, Jacobsthal and Mersenne numbers. Their Binet’s formulas are obtained. We also define the matrices of the Gauss Lichtenberg numbers. We examine properties of the matrices.

Keywords

Gauss numbers Jacobsthal numbers Lichtenberg numbers Mersenne numbers

References

  • [1] Y. Alp, E. G. Kocer, Hybrid Leonardo numbers, Chaos Solitons Fractals 150, Article ID 111128 (2021).
  • [2] M. As¸ci, E. G¨urel, Gaussian Jacobsthal and Gaussian Jacobsthal–Lucas numbers, Ars. Comb. 111 (2013), 53–63.
  • [3] M. As¸ci, E. G¨urel, Gaussian Jacobsthal and Gaussian Jacobsthal–Lucas polynomials, Notes on Number Theory and Discrete Mathematics 19(1) (2013), 25–36.
  • [4] P. Catarino, H. Campos, P. Vasco, On the Mersenne sequence, Annales Mathematicae et Informaticae 46 (2016), 37–53.
  • [5] A. Das¸demir, G. Bilgici, Gaussian Mersenne numbers and generalized Mersenne quaternions, Notes on Number Theory and Discrete Mathematics 25(3) (2019), 87–96.
  • [6] G. B. Djordjevi´c, Generalized Jacobsthal polynomials, Fibonacci Quart. 38 (2000), 239–243.
  • [7] G. B. Djordjevi´c, H. M. Srivastava, Incomplete generalized Jacobsthal and Jacobsthal–Lucas numbers, Mathematical and Computer Modelling 42(9) (2005), 1049–1056.
  • [8] A. M. Hinz, The Lichtenberg sequence, Fibonacci Quart. 55(2) (2017), 2–12.
  • [9] A. F. Horadam, Jacobsthal representation polynomials, Fibonacci Quart. 35 (1997), 137–148.
  • [10] A. F. Horadam, Jacobsthal representation numbers, The Fibonacci Quarterly 43(1) (1996), 40–54.
  • [11] A. F. Horadam, A generalized Fibonacci sequence, Am. Math. Mon. 68(5) (1961), 455–459.
  • [12] J. H. Jordan, Gaussian Fibonacci and Lucas numbers, Fibonacci Quart. 3(4) (1965), 315–318.
  • [13] C. Kızılates¸, New families of Horadam numbers associated with finite operators and their applications, Math. Methods Appl. Sci. 44(18) (2021), 14371–14381.
  • [14] C. Kızılates¸, N. Tuglu, N., B. C¸ ekim, Binomial transforms of of Quadrapell sequences and Quadrapell matrix sequences, Journal of Science and Arts 1(38) (2017), 69–80.
  • [15] C. Kızılates¸, W. S. Du, F. Qi, Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences, Tamkang Journal of Mathematics 53(3) (2022), 277–291.
  • [16] C. Kızılates¸, P. Catarino, N. Tuglu, On the bicomplex generalized Tribonacci quaternions, Mathematics 7:80 (2019).
  • [17] M. Kumari, K. Prasad, J. Tanti, On the generalization of Mersenne and Gaussian Mersenne polynomials, J Anal. 32 (2024), 931–947.
  • [18] G. Morales, On Gauss third-order Jacobsthal numbers and their applications, An. S¸ tiint¸. Univ. Al. I. Cuza Ias¸i. Mat. (N.S.) Tomul LXVII, f. 2 (2021), 231–241.
  • [19] G. Morales, Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal-Lucas polynomials and their properties, Asian-European Journal of Mathematics 14(5) (2021), 1–12.
  • [20] G. Morales, Identities for third order Jacobsthal quaternions, Adv. Appl. Clifford Algebra 27(2) (2017), 1043–1053.
  • [21] G. Morales, On a generalization of Tribonacci quaternions, Mediter. J. Math. 14:239 (2017).
  • [22] Morales, G. Binomial transforms of the third-order Jacobsthal and modified third-order Jacobsthal polynomials, Univers. J. Math. Appl. 7(3) (2024), 144–151.
  • [23] P. K. Stockmeyer, An exploration of sequence A000975, Fibonacci Quart. 55(5) (2017), 174–185.

Year 2025, Volume: 13 Issue: 1, 21 - 27, 30.04.2025

Gamaliel Morales

Abstract

References

  • [1] Y. Alp, E. G. Kocer, Hybrid Leonardo numbers, Chaos Solitons Fractals 150, Article ID 111128 (2021).
  • [2] M. As¸ci, E. G¨urel, Gaussian Jacobsthal and Gaussian Jacobsthal–Lucas numbers, Ars. Comb. 111 (2013), 53–63.
  • [3] M. As¸ci, E. G¨urel, Gaussian Jacobsthal and Gaussian Jacobsthal–Lucas polynomials, Notes on Number Theory and Discrete Mathematics 19(1) (2013), 25–36.
  • [4] P. Catarino, H. Campos, P. Vasco, On the Mersenne sequence, Annales Mathematicae et Informaticae 46 (2016), 37–53.
  • [5] A. Das¸demir, G. Bilgici, Gaussian Mersenne numbers and generalized Mersenne quaternions, Notes on Number Theory and Discrete Mathematics 25(3) (2019), 87–96.
  • [6] G. B. Djordjevi´c, Generalized Jacobsthal polynomials, Fibonacci Quart. 38 (2000), 239–243.
  • [7] G. B. Djordjevi´c, H. M. Srivastava, Incomplete generalized Jacobsthal and Jacobsthal–Lucas numbers, Mathematical and Computer Modelling 42(9) (2005), 1049–1056.
  • [8] A. M. Hinz, The Lichtenberg sequence, Fibonacci Quart. 55(2) (2017), 2–12.
  • [9] A. F. Horadam, Jacobsthal representation polynomials, Fibonacci Quart. 35 (1997), 137–148.
  • [10] A. F. Horadam, Jacobsthal representation numbers, The Fibonacci Quarterly 43(1) (1996), 40–54.
  • [11] A. F. Horadam, A generalized Fibonacci sequence, Am. Math. Mon. 68(5) (1961), 455–459.
  • [12] J. H. Jordan, Gaussian Fibonacci and Lucas numbers, Fibonacci Quart. 3(4) (1965), 315–318.
  • [13] C. Kızılates¸, New families of Horadam numbers associated with finite operators and their applications, Math. Methods Appl. Sci. 44(18) (2021), 14371–14381.
  • [14] C. Kızılates¸, N. Tuglu, N., B. C¸ ekim, Binomial transforms of of Quadrapell sequences and Quadrapell matrix sequences, Journal of Science and Arts 1(38) (2017), 69–80.
  • [15] C. Kızılates¸, W. S. Du, F. Qi, Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences, Tamkang Journal of Mathematics 53(3) (2022), 277–291.
  • [16] C. Kızılates¸, P. Catarino, N. Tuglu, On the bicomplex generalized Tribonacci quaternions, Mathematics 7:80 (2019).
  • [17] M. Kumari, K. Prasad, J. Tanti, On the generalization of Mersenne and Gaussian Mersenne polynomials, J Anal. 32 (2024), 931–947.
  • [18] G. Morales, On Gauss third-order Jacobsthal numbers and their applications, An. S¸ tiint¸. Univ. Al. I. Cuza Ias¸i. Mat. (N.S.) Tomul LXVII, f. 2 (2021), 231–241.
  • [19] G. Morales, Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal-Lucas polynomials and their properties, Asian-European Journal of Mathematics 14(5) (2021), 1–12.
  • [20] G. Morales, Identities for third order Jacobsthal quaternions, Adv. Appl. Clifford Algebra 27(2) (2017), 1043–1053.
  • [21] G. Morales, On a generalization of Tribonacci quaternions, Mediter. J. Math. 14:239 (2017).
  • [22] Morales, G. Binomial transforms of the third-order Jacobsthal and modified third-order Jacobsthal polynomials, Univers. J. Math. Appl. 7(3) (2024), 144–151.
  • [23] P. K. Stockmeyer, An exploration of sequence A000975, Fibonacci Quart. 55(5) (2017), 174–185.
There are 23 citations in total.

Details

Primary Language English
Subjects Complex Systems in Mathematics
Journal Section Articles
Authors

Gamaliel Morales 0000-0003-3164-4434

Early Pub Date April 28, 2025
Publication Date April 30, 2025
Submission Date July 22, 2024
Acceptance Date November 5, 2024
Published in Issue Year 2025 Volume: 13 Issue: 1

Cite

APA Morales, G. (2025). Some Properties between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers. Konuralp Journal of Mathematics, 13(1), 21-27.
AMA Morales G. Some Properties between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers. Konuralp J. Math. April 2025;13(1):21-27.
Chicago Morales, Gamaliel. “Some Properties Between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers”. Konuralp Journal of Mathematics 13, no. 1 (April 2025): 21-27.
EndNote Morales G (April 1, 2025) Some Properties between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers. Konuralp Journal of Mathematics 13 1 21–27.
IEEE G. Morales, “Some Properties between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers”, Konuralp J. Math., vol. 13, no. 1, pp. 21–27, 2025.
ISNAD Morales, Gamaliel. “Some Properties Between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers”. Konuralp Journal of Mathematics 13/1 (April 2025), 21-27.
JAMA Morales G. Some Properties between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers. Konuralp J. Math. 2025;13:21–27.
MLA Morales, Gamaliel. “Some Properties Between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers”. Konuralp Journal of Mathematics, vol. 13, no. 1, 2025, pp. 21-27.
Vancouver Morales G. Some Properties between Lichtenberg, Jacobsthal and Mersenne Gaussian Numbers. Konuralp J. Math. 2025;13(1):21-7.
 Download Cover Image
Creative Commons License
The published articles in KJM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.