Hybrid Quaternions of Leonardo

Mangueira, M. C. S.; Alves, Francisco Regis Vieira; Catarino, Paula

doi:10.5540/tcam.2022.023.01.00051

Cited by 6 publications

(7 citation statements)

References 18 publications

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“…In Mangueira et al (2022a), a study was conducted on the complexification of the Leonardo numbers, portraying their quaternions (numbers presented in four dimensions). Mangueira et al (2022a) associated quaternions with the Leonardo hybrids, resulting in the Leonardo hybrid quaternions, presenting their recurrence, Binet's formula, generating function, and identities related to this association.…”

Section: A State Of Art On the Leonardo Sequence: Panorama Of Current...mentioning

confidence: 99%

State of the art on the Leonardo sequence: An evolutionary study of the epistemic-mathematical field

Mangueira,

Alves,

Catarino

et al. 2024

PEDAGOGICAL RES

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This work is a segment of an ongoing doctoral research in Brazil. The Leonardo numbers and the Leonardo sequence have gained attention from mathematicians and the academic community. Despite being a relatively new sequence within mathematical literature, its discussion has intensified over the past five years, giving rise to other branches, with contributions and associations to other topics in mathematics. Thus, the aim of this study was to construct and present the state of the art of the Leonardo sequence, considering its historical aspects and highlighting works on its evolutionary process in the epistemic-mathematical field, regarding its generalization, complexification, hyper complexification, and combinatorial model during the last five years (2019-2023). The methodology used was a bibliographic study, where the state of the art was carried out through the mapping of publications on the subject. Twenty-four research works related to the key descriptors “Leonardo sequence”, “Leonardo numbers”, “complexification”, “generalization”, “hybrids”, and “combinatorial model” were found, cataloged, and discussed. From the analysis of these studies, it is noted that its development in pure mathematics has advanced to other branches and discoveries, and that, albeit timidly, research on the subject has emerged directed towards the field of education, especially in the initial teacher training and, particularly, in Brazil.

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Section: A State Of Art On the Leonardo Sequence: Panorama Of Current...mentioning

confidence: 99%

State of the art on the Leonardo sequence: An evolutionary study of the epistemic-mathematical field

Mangueira,

Alves,

Catarino

et al. 2024

PEDAGOGICAL RES

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“…Many authors have extensively researched different types of quaternions and hybrid numbers, where their components are derived from terms found in special integer sequences. In particular, Leonardo hybrid quaternions were studied in [24], Leonardo sedenions were studied in [25], Szynal [26] studied Horadam hybrid numbers, which generalize the classical Fibonacci hybrid numbers and the classical Lucas hybrid numbers. Polynomial versions of Fibonacci and Lucas hybrid numbers were studied in [27].…”

Section: Introductionmentioning

confidence: 99%

A New Class of Leonardo Hybrid Numbers and Some Remarks on Leonardo Quaternions over Finite Fields

Tan,

Savin,

Yılmaz

2023

Mathematics

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In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Zp for special values of prime integer p.

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“…Hybrid numbers have been the subject of much research recently [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Szynal-Liana and Wloch [16] defined the 𝑛-th Fibonacci hybrid number as…”

Section: Introductionmentioning

confidence: 99%

“…The Horadam hybrid quaternions and some special classes of number sequences such as Fibonacci, Lucas, Pell and Jacobsthal hybrid quaternions are introduced by Dağdeviren and Kürüz [29]. Mangueira et al [30] defined Leonardo quaternions and Leonardo hybrid quaternions. The authors also presented the recurrence relation, characteristic equation, generating function, Binet formula for the Leonardo hybrid quaternions and its relations with the Fibonacci quaternions.…”

Section: Introductionmentioning

confidence: 99%