Gaussian Mersenne numbers and generalized Mersenne quaternions

Ahmet Daşdemir and Göksal Bilgici
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 3, Pages 87-96
DOI: 10.7546/nntdm.2019.25.3.87-96
Full paper (PDF, 156 Kb)

Details

Authors and affiliations

Ahmet Daşdemir
Department of Mathematics
Faculty of Arts and Sciences
Kastamonu University
Kuzeykent Campus, 37150 Kastamonu, Turkey

Göksal Bilgici
Department of the Computer Education and Instructional Technologies
Education Faculty
Kastamonu University
37100, Kastamonu, Turkey

Abstract

In this study, we introduce a new class of quaternions associated with the well-known Mersenne numbers. There are many studies about the quaternions with special integer sequences and their generalizations. All of these studies used consecutive elements of the considered sequences. Here, we extend the usual definitions into a wider structure by using arbitrary Mersenne numbers. Moreover, we present Gaussian Mersenne numbers. In addition, we give some properties of this type of quaternions and Gaussian Mersenne numbers, including generating function and Binet-like formula.

Keywords

• Mersenne quaternions,
• Generating function,
• Binet’s formula,
• Gaussian Mersenne numbers
• Catalan’s identity

2010 Mathematics Subject Classification

• 11B37
• 11B39

References

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