A new kind of Mersenne numbers: the Tricomplex Mersenne sequence

Fernando Soares de Carvalho; Paula M. M. C. Catarino; Helena Campos; Eudes Antonio Costa

doi:10.14244/lajm.v5i1.92

Authors

Fernando Soares de Carvalho Universidade Federal do Tocantins https://orcid.org/0000-0001-6639-0716
Paula M. M. C. Catarino University of Trás-os-Montes and Alto Douro https://orcid.org/0000-0001-6917-5093
Helena Campos University of Trás-os-Montes and Alto Douro https://orcid.org/0000-0003-2767-0998
Eudes Antonio Costa Universidade Federal do Tocantins https://orcid.org/0000-0001-6684-9961

DOI:

https://doi.org/10.14244/lajm.v5i1.92

Keywords:

Binet formula, Mersenne-type sequences, Tricomplex Mersenne sequences, Generating function

Abstract

In this paper, a new family of arithmetic sequences associated with Mersenne numbers is introduced, and their properties are explored. The classical Mersenne sequence \(\{M_n\}_{n \geq 0}\) is defined by the initial values \(M_0 = 0\) and \(M_1 = 1\), with the recurrence relation \(M_n = 3M_{n-1} -2 M_{n-2}\) for \(n \geq 2\). This concept is extended by defining the tricomplex Mersenne numbers within the tricomplex ring \((\mathbb{T}, +, \times)\). A homogeneous recurrence relation for these new numbers is presented, and their various mathematical \mbox{properties} are studied.
The tricomplex Mersenne sequence \(\{TM^{\ast}_n\}_{n \geq 0}\) is defined by the initial \mbox{values} \(TM^{\ast}_0\) and \(TM^{\ast}_1\), as well as the same Mersenne recurrence relation. Key formulas, such as the Binet formula, are derived, and fundamental \mbox{identities} like those of Tagiuri-Vajda, d'Ocagne, and Catalan are explored. In addition, generating functions are presented, and novel identities related to the Mersenne numbers are discovered through this extended framework. Several properties of the tricomplex Mersenne sequence are also established, and results involving sums of terms are provided. These findings contribute to a deeper understanding of generalized Mersenne-type sequences and their potential applications.

Author Biographies

Fernando Soares de Carvalho, Universidade Federal do Tocantins

Associate Professor at the Federal University of Tocantins, Brazil. His research interests include Number Theory, Discrete Mathematics, and Applied Mathematics.

Paula M. M. C. Catarino, University of Trás-os-Montes and Alto Douro

PAULA CATARINO has a PhD in Mathematics. Full Professor at the Department of Mathematics at the University of Trás-os-Montes and Alto Douro (UTAD). Researcher at the Research Center CMAT-UTAD- CMAT Pole of the University of Minho and also Researcher at the Research Center CIDTFF - Research Center “Didactics and Technology in Trainer Training. Author of several scientific articles published in international peer-reviewed journals.

Present research interests:
The current research around the sequences of integers inspires and provide numerous specializations and generalizations. From 2013 she studied sequences of special numbers, investigating some of their algebraic properties, their generating functions and generating matrices. Sequences of quaternions, octonions, sedenions and polynomials will also be the focus of her research work. The relationship of these sequences with several types of matrices (for instance, Hessenberg, circulant, tridiagonal matrices, etc.), whose entries are the elements of these sequences, and respective norms will be studied, providing some results involving them and study the invertibility of these type matrices.
Also she is working on semigroups and in area of Mathematical Education.
Institutional address
Dep. Mathematics – Escola de Ciências e Tecnologia - Universidade de Trás-os-Montes e Alto Douro (UTAD)
Quinta de Prados; 5000-801 Vila Real; PORTUGAL
Telephone: +351 259350818 | Email: pcatarin@utad.pt
Academic degrees
BSc in Geographical Engineering (1984); MSc In Mathematics-Algebra (1991); PhD in Mathematics (1998).

Helena Campos, University of Trás-os-Montes and Alto Douro

Helena Campos holds a degree in Pure Mathematics and another one in Mathematical Education, both from Coimbra University.
After being a mathematics teacher of Basic and Secondary Education for a few years, in 1995 she joined the Trás-os-Montes e Alto Douro University (UTAD).
She obtained a master’s degree in Mathematics Education from Minho University (Braga) in the year 2000.
Her PhD in Mathematics, with a Specialisation in Topology and Geometry, was obtained from UNED in Madrid (Spain) in 2008.
She was a Research member of The Research Centre of Mathematics - CM-UTAD - (UTAD) from July 2007 to October 2015.
Since December 2015 she has been a Research member of the Research Centre of Didactics and Technology in the Education of Trainers at CIDTFF (Aveiro University).
She is also an Assistant Professor in the Mathematics Department at UTAD.
Her research interests include the fields: Didactics of Mathematics, mainly in the Didactics of Geometry, Dynamic Geometry Environments in Basic and Secondary Education and Mathematical Creativity in Geometric Reasoning Processes; Teaching and learning Mathematics in the first years of school (Pre-School and Basic Education); Teacher Training; and Geometric connections with Theory of numbers.

Eudes Antonio Costa, Universidade Federal do Tocantins

Adjunct Professor at the Federal University of Tocantins, Arraias Campus (Mathematics Course). Post-doctorate in Mathematics from the Federal University of Ceará (2019), PhD in Mathematics from the University of Brasília (2013), Master's degree in Mathematics from the Federal University of Goiás (2001), undergraduate degree in Mathematics from the Federal University of Goiás (1998) and undergraduate degree in Philosophy from the Pontifical Catholic University of Goiás (1995). Experience with Teacher Training (PROFMAT, Bachelor's Degree Course and Improvement Courses) and Mathematics Olympiads (OBM and OBMEP).

References

[1] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, OEIS Foundation Inc., 2026.

[2] P. Catarino, H. Campos and P. Vasco, On the Mersenne sequence, Annales Mathematicae et Informaticae 46 (2016), 37–53.

[3] T. Goy, On new identities for Mersenne numbers, Applied Mathematics E-Notes 18 (2018), 100–105.

[4] B. A. Delello, Perfect Numbers and Mersenne Primes, PhD Thesis, University of Central Florida, 1986.

[5] E. Deza, Mersenne Numbers and Fermat Numbers, Vol. 1, World Scientific, Singapore, 2021. DOI: https://doi.org/10.1142/12100

[6] J. S. Hall, A remark on the primeness of Mersenne numbers, Journal of the London Mathematical Society 1 (1953), no. 3, 285–287. DOI: https://doi.org/10.1112/jlms/s1-28.3.285

[7] T. Koshy, Catalan Numbers with Applications [Chapter 13: Divisibility Properties], Oxford University Press, New York, 2009. DOI: https://doi.org/10.1093/acprof:oso/9780195334548.003.0013

[8] D. C. Santos, E. A. Costa and P. M. M. C. Catarino, On Gersenne sequence: A study of one family in the Horadam-type sequence, Axioms 14 (2025), no. 3. DOI: https://doi.org/10.3390/axioms14030203

[9] K. Mondal and S. Pramanik, Tri-complex rough neutrosophic similarity measure and its application in multi-attribute decision making, Critical Review 11 (2015), 26–40.

[10] S. Olariu, Complex Numbers in n Dimensions, Elsevier Science B. V., Amsterdam, 2002.

[11] S. Olariu, Complex numbers in three dimensions, arXiv preprint math/0008120 (2000).

[12] A. Ottoni, N. C. L. Deus and J. E. Ottoni, A Álgebra dos números ternários, Revista de Matemática da UFOP 1 (2024).

[13] E. A. Costa, P. M. M. C. Catarino, F. S. Monteiro, V. M. A. Souza and D. C. Santos, Tricomplex Fibonacci numbers: A new family of Fibonacci-type sequences, Mathematics 12 (2024), no. 23, Article 3723. DOI: https://doi.org/10.3390/math12233723

[14] E. A. Costa, P. M. M. C. Catarino and D. C. Santos, A study of the symmetry of the tricomplex repunit sequence with repunit sequence, Symmetry 17 (2025), no. 1. DOI: https://doi.org/10.3390/sym17010028

[15] D. C. Santos, P. M. M. C. Catarino and E. A. Costa, On Tricomplex Horadam numbers: A new class of Horadam sequences, Symmetry 17 (2025), no. 10. DOI: https://doi.org/10.3390/sym17101616

[16] E. V. P. Spreafico, P. M. Catarino and E. A. Costa, A note on tricomplex Leonardo numbers, Mathematica Bohemica [first online], (2026). DOI: https://doi.org/10.21136/MB.2026.0137-25

[17] J. K. Baumgart, Tópicos de História da Matemática para Uso em Sala de Aula: Álgebra, Atual Editora, São Paulo, 1992.

[18] C. B. Boyer and U. C. Merzbach, História da Matemática, Editora Blucher, São Paulo, 2019.

[19] H. Eves, Introdução à História da Matemática, Editora da UNICAMP, São Paulo, 2008.

[20] B. Felzenszwalb, Álgebras de Dimensão Finitas, IMPA–CNPq, Rio de Janeiro, 1979.

[21] W.-D. Richter, On complex numbers in higher dimensions, Axioms 11 (2022), no. 1. DOI: https://doi.org/10.3390/axioms11010022

[22] T. Roque, História da Matemática, Editora Schwarcz–Companhia das Letras, São Paulo, 2012.

[23] M. Chelgham and A. Boussayoud, On the k-Mersenne Lucas numbers, Notes on Number Theory and Discrete Mathematics 27 (2021), no. 1, 7–13. DOI: https://doi.org/10.7546/nntdm.2021.27.1.7-13

[24] N. Saba and A. Boussayoud, On the bivariate Mersenne Lucas polynomials and their properties, Chaos, Solitons & Fractals 146 (2021), Article 110899. DOI: https://doi.org/10.1016/j.chaos.2021.110899

[25] N. Saba, A. Boussayoud and K. V. V. Kanuri, Mersenne Lucas numbers and complete homogeneous symmetric functions, Journal of Mathematics and Computer Science 24 (2021), no. 2, 127–139. DOI: https://doi.org/10.22436/jmcs.024.02.04

[26] A. F. Horadam, Basic properties of a certain generalized sequence of numbers, The Fibonacci Quarterly 3 (1965), no. 3, 161–176. DOI: https://doi.org/10.1080/00150517.1965.12431416

[27] D. Kalman and R. Mena, The Fibonacci numbers—exposed, Mathematics Magazine 76 (2003), no. 3, 167–181. DOI: https://doi.org/10.1080/0025570X.2003.11953176

[28] Y. Soykan, A study on generalized Mersenne numbers, Journal of Progressive Research in Mathematics 18 (2021), no. 3, 90–112.

[29] G. Cerda, Matrix methods in Horadam sequences, Boletín de Matemáticas 19 (2012), no. 2, 97–106.