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

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## 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

- Berzsenyi, G. (1977). Gaussian Fibonacci numbers, Fibonacci Quart., 15, 233–236.
- Catarino, P., Campos, H., & Vasco, P. (2016). On the Mersenne sequence, Ann. Math. Inform., 46, 37–53.
- Cimen, C. B., & Ipek, A. (2016). On Pell quaternions and Pell–Lucas quaternions, Adv. Appl. Clifford Algebr., 26, 39–51.
- Hamilton, W. R. (1853). Lectures on Quaternions, Hodges and Smith, Dublin.
- Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly., 70, 289–291.
- Szynal-Liana, A., & Wloch, I. (2016). The Pell quaternions and the Pell octonions, Adv. Appl. Clifford Algebr., 26, 435–440.
- Szynal-Liana, A., & Wloch, I. (2016). A note on Jacobsthal quaternions, Adv. Appl. Clifford Algebr., 26, 441–447.
- Wright, G. H., & Hardy, E. M. (1975). An Introduction to the Theory of Numbers, Oxford University Press, Oxford.
- Zeilberger, D. (1991). The method of creative telescoping, J. Symbolic Comput., 11, 195–204.

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## Cite this paper

APADaşdemir, A. & Bilgici, G. (2019). Gaussian Mersenne numbers and generalized Mersenne quaternions. Notes on Number Theory and Discrete Mathematics, 25(3), 87-96, doi: 10.7546/nntdm.2019.25.3.87-96.

ChicagoDaşdemir, A. & Bilgici, G. (2019). “SGaussian Mersenne numbers and generalized Mersenne quaternions.” Notes on Number Theory and Discrete Mathematics. Notes on Number Theory and Discrete Mathematics 25, no. 3 (2019): 87-96, doi: 10.7546/nntdm.2019.25.3.87-96.

MLADaşdemir, A. & Bilgici, G. (2019). “Gaussian Mersenne numbers and generalized Mersenne quaternions.” Notes on Number Theory and Discrete Mathematics 25.3 (2019): 87-96. Print, doi: 10.7546/nntdm.2019.25.3.87-96.