Split Pell and Pell–Lucas Quaternions

Tokeşer, Ümit; Ünal, Zafer; Bilgici, Göksal

doi:10.1007/s00006-016-0747-x

Cited by 23 publications

(14 citation statements)

References 14 publications

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“…The reader shall see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] for some works on number theory and its applications.…”

Section: Resultsmentioning

confidence: 99%

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Mamombe¹

2019

JAMCS

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The proportiones perfectus law states that σ_x^y=(x+√(x^2+4y))/2 is a proportione perfectus if 1≤y≤x such that for an arbitrary positive integer , there exists an integer sequence defined simultaneously by the quasigeometric relation h_(n+1)=round(σ_x^y h_n ),n≥1 and the arithmetic relation h_(n+2)=〖xh〗_(n+1)+〖yh〗_n,n≥1. When x=y=1 σ_x^y is the golden mean. When x=2,y=1,σ_x^y is the silver mean. In previous works we introduced the theory of number genetics – a framework of logic within which the golden section is studied. In this work we apply the concept to all proportiones perfectus. Let be defined as above. Furthermore, let h1 satisfy {█(round(h_1/(σ_x^y ))=w@round(wσ_x^y)≠h_1 ). ┤Now let H'nbe Hn with h1=1 Again let be defined as A robust universal computing machine is herein developed for the purpose of establishing the relationship h_i=g_(n+i-1)±h_i^',i,n≥1, which relationship is key to the logic protocol. It is clear therefore that this system of logic presents as the building block of every (including itself) within a particular σ_x^y regime. Clearly number genetics takes central place in the proportiones perfectus theory.

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“…The reader shall see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] for some works on number theory and its applications.…”

Section: Resultsmentioning

confidence: 99%

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Mamombe¹

2019

JAMCS

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“…First the idea to consider Pell quaternions was suggested by Horadam in paper [13]. In the literature, the reader can find Pell quaternions and studies on their properties in [1,2,4,8,9,11,18,19].…”

Section: Introductionmentioning

confidence: 99%

Dual-complex k-Pell quaternions

Aydın¹

2019

NNTDM

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In this paper, dual-complex k-Pell numbers and dual-complex k-Pell quaternions are defined. Also, some algebraic properties of dual-complex k-Pell numbers and quaternions which are connected with dual-complex numbers and k-Pell numbers are investigated. Furthermore, Honsberger Identity, d'Ocagne's Identity, Binet's Formula, Cassini's Identity and Catalan's Identity for these quaternions are given.

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“…The quaternion version of Pell sequence and some generalizations have been considered by several researchers in their works (see, for example, [4,7,21,23]).…”

Section: Introductionmentioning

confidence: 99%

Bicomplex k-Pell Quaternions

Catarino

2018

Comput. Methods Funct. Theory

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Split Pell and Pell–Lucas Quaternions

Cited by 23 publications

References 14 publications

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Dual-complex k-Pell quaternions

Bicomplex k-Pell Quaternions

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