Two-dimensional, three-dimensional and n-dimensional Mersenne relations

Authors

DOI:

https://doi.org/10.14393/BEJOM-v4-2023-69300

Keywords:

Mersenne Sequence, Two-dimensional relationships, Three-dimensional relationships, n-dimensional relations

Abstract

The Mersenne sequence, named in honor of the French mathematician Marin Mersenne, is a second-order recursive progression represented by a one-dimensional model, and its numbers can also be described in the form Mn=2n-1. This research aims to explore and investigate the recurring relationships in two-dimensional (M(n,m)), three-dimensional (M(n,m,p)), and n-dimensional (M(n1, n2, n3, ... ,nt)) spaces based on the one-dimensional model Mn+1 = 3Mn - 2Mn-1, where n is a non-negative integer, and the initial values are set as M0 = 0 and M1 = 1. The evolution of the sequence is examined along with its process of complexification. Throughout this analysis, mathematical properties of these number relations are identified, emphasizing the dimensional expansion of the sequence and the inclusion of imaginary units i, j, ... , µn, culminating in the generalization of n-dimensional relationships.

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Author Biographies

Milena Carolina dos Santos Mangueira, Instituto Federal de Educação, Ciência e Tecnologia do Estado do Ceará - IFCE

Has a bachelor's degree in Mathematics from the State University of Rio Grande do Norte (2018), a master's degree in Science and Mathematics Education from the Federal Institute of Education, Science and Technology of Ceará (2022), and an ongoing doctoral program in Education through the RENOEN network at the Federal Institute of Education, Science and Technology of Ceará, funded by the National Council for Scientific and Technological Development – CNPq.

Renata Passos Machado Vieira, Secretaria de Educação do Estado do Ceará - SEDUC

Master's degree in Science and Mathematics Education from the Federal Institute of Education, Science and Technology of the State of Ceará (IFCE). Specialized in School Management and Pedagogical Practices from Candido Mendes University (RJ) (2016), graduated in Telecommunications Engineering from the Federal Institute of Education, Science and Technology of Ceará (2013), and in Mathematics Education from the Integrated College of Greater Fortaleza (2014). Currently a teacher at the State Department of Education of Ceará (SEDUC). Ongoing Ph.D. in Education through the RENOEN network at the Federal University of Ceará (UFC), funded by the Foundation for Support of Scientific and Technological Development of Ceará (Funcap).

Francisco Regis Vieira Alves, Instituto Federal de Educação Ciência e Tecnologia do estado do Ceará - IFCE

Has a bachelor's degree in Mathematics from the Federal University of Ceará (1998), a bachelor's degree in Mathematics Education from the Federal University of Ceará (1997), a master's degree in Pure Mathematics from the Federal University of Ceará (2001), and a master's degree in Education with a focus on Mathematics Education from the Federal University of Ceará (2002). Ph.D. with an emphasis on Mathematics Education (UFC - 2011). Currently a FULL Professor at the Federal Institute of Education, Science and Technology of the state of Ceará/IFCE and a CNPq Level 2 Productivity Fellow. Professor of the Doctorate in the Networked Graduate Program in Teaching (RENOEN) and the Academic Master's in Science and Mathematics Education of the Professional Master's in Professional and Technological Education PROEPT-IFCE. Editor-in-chief of the journal "Ensino em Debate" (REDE), the official journal linked to the Doctorate in Teaching - Northeast Teaching Network (RENOEN) and the Master's Program in Science and Mathematics Education (PGECM/IFCE).

Paula Maria Machado Cruz Catarino, Universidade de Trás- os-Montes e Alto Douro - UTAD

Bachelor's degree in Geographical Engineering (1984); Master's degree in Mathematics- Algebra (1991). Ph.D. in Mathematics. Full Professor at UTAD (University of Trás-os- Montes and Alto Douro) in Portugal. Researcher at the CMAT-UTAD Research Center - CMAT Pole at the University of Minho and also a researcher at the CIDTFF Research Center - "Didactics and Technology in Teacher Training." Currently a member of the General Council of UTAD.

Roger Oliveira Sousa, Faculdade de Educação, Ciências e Letras do Sertão Central - FECLESC/UECE

Has a bachelor's degree in Mathematics Education from the Federal Institute of Ceará and a master's degree in Mathematics from the Federal University of Ceará (2014). Ph.D. in Mathematics from the Federal University of Ceará. Professor at FECLESC/UECE - Faculty of Education, Sciences, and Letters of the Central Sertão.

References

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HARMAN, C. J. Complex Fibonacci numbers. The Fibonacci Quarterly, v. 19, n.1, p. 82-86, 1981.

KOSHY, T.; GAO, Z. Catalan numbers with Mersenne subscripts. Math. Scientist, v. 38, n. 2, p. 86-91, 2013.

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OLIVEIRA, R. R. de; ALVES, F. R. V.; PAIVA, R. E. B. Identidades bi e tridimensionais para os números de Fibonacci na forma complexa. C.Q.D.-Revista Eletrônica Paulista de Matemática, Bauru, v. 11, n. 2, p. 91-106, 2017.

VIEIRA, R. P. M.; ALVES, F. R. V.; CATARINO, P. M. M. C. Relações bidimensionais e identidades da sequência de Leonardo. Revista Sergipana de Matemática e Educação Matemática, v. 4, n. 2, p. 156-173, 2019.

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Published

2023-12-21

How to Cite

MANGUEIRA, M. C. dos S.; VIEIRA, R. P. M.; ALVES, F. R. V.; CATARINO, P. M. M. C.; SOUSA, R. O. Two-dimensional, three-dimensional and n-dimensional Mersenne relations. BRAZILIAN ELECTRONIC JOURNAL OF MATHEMATICS, Uberlândia, v. 4, p. 1–29, 2023. DOI: 10.14393/BEJOM-v4-2023-69300. Disponível em: https://seer.ufu.br/index.php/BEJOM/article/view/69300. Acesso em: 23 nov. 2024.

Issue

Vol. 4 (2023): Brazilian Electronic Journal of Mathematics

Section

Articles - Pure Mathematics

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Copyright (c) 2023 Milena Carolina dos Santos Mangueira, Renata Passos Machado Vieira, Francisco Régis Vieira Alves, Paula Maria Machado Cruz Catarino, Roger Oliveira Sousa Sousa

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