S. G. Rayaguru, D. Savin and G. K. Panda

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 2, Pages 91-112

DOI: 10.7546/nntdm.2019.25.2.91-112

**Full paper (PDF, 249 Kb)**

## Details

### Authors and affiliations

S. G. Rayaguru

*Department of Mathematics, National Institute of Technology
Rourkela, India
*

D. Savin

*Ovidius University
Bd. Mamaia 124, 900527 Constanta, Romania
*

G. K. Panda

*Department of Mathematics, National Institute of Technology
Rourkela, India
*

### Abstract

Fibonacci and Lucas symbol elements are generalized to Horadam symbol elements and some properties are studied. In the last section we use these properties for to find zero divisors in symbol algebras over cyclotomic fields of finite fields.

### Keywords

- Recurrence relations
- Quaternions
- Symbol algebras

### 2010 Mathematics Subject Classification

- 11R52
- 11B37
- 11B83

### References

- Akyuz, Z., & Halici, S. (2013). On binomial sums for the general second order linear recurrence,Hacet. J. Math. Stat., 42 (4), 431–435.
- Bolat, C., & Kose, H. (2010). On the Properties of
*k*-Fibonacci Numbers, Int. J. Contemp. Math. Sciences, 5 (22), 1097–1105. - Cerda-Morales, G., & Kose, H. (2017). Some Properties of Horadam Quaternions, Available online: https://arxiv.org/pdf/1707.05918.pdf.
- Flaut, C., & Savin, D. (2014). Some properties of symbol algebras of degree three, Math. Reports, 16 (66), 3, 443–463.
- Flaut, C., & Shpakivskyi, V. (2013). On generalized Fibonacci quaternions and Fibonacci–Narayana quaternions, Adv. Appl. Clifford Algebras, 23, 673–688.
- Flaut, C., Savin, D., & Iorgulescu, G. (2013). Some properties of Fibonacci and Lucas symbol elements, J. Math. Sci. Adv. Appl, 20, 37–43.
- Flaut, C., & Savin, D. (2017).Some remarks regarding(a,b,x0,x1)-numbers and (
*a*,*b*,*x*_{0},*x*_{1})-quaternions, accepted in Ars Combinatoria (Available online: https://arxiv.org/pdf/1705.00361.pdf). - Flaut, C., & Savin, D. (2018). Some special number sequences obtained from a difference equation of degree three, Chaos Solitons Fractals, 106, 67–71.
- Halici, S. (2012). On Fibonacci quaternions, Adv. Appl. Clifford Algebr., 22 (2), 321–327.
- Halici, S. (2013). On Complex Fibonacci quaternions, Adv. Appl. Clifford Algebr., 23, 105–112.
- Halici, S., & Karatas ̧A. (2017). On a generalization for Fibonacci quaternions, Chaos Solitons Fractals, 98, 178–182.
- Hazewinkel, M. (2000). Handbook of Algebra, Vol. 2, Amsterdam, North Holland, 39.
- Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70, 289–291.
- Ipek, A. (2017) On (
*p*,*q*)-Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums, Adv. Appl. Clifford Algebr., 27 (2), 1343–1351. - Karatas ̧A., & Halici, S. (2017). Horadam Octonions, An. St. Univ. Ovidius Constanta, Mat. Ser., 25 (3), 97–106.
- Kilic, E. (2010). On binomial sums for the general second order linear recurrence, Integers, 10, 801–806.
- Pierce, R. S. (1982). Associative Algebras, Springer Verlag, New York, Heidelberg, Berlin.
- Savin, D. (2015). Some properties of Fibonacci numbers, Fibonacci octonions and generalized Fibonacci–Lucas octonions, Advances in Difference Equations, 2015:298, 1–10, DOI 10.1186/s13662-015-0627-z.
- Savin, D. (2017). About Special Elements in Quaternion Algebras Over Finite Fields, Adv. Appl. Clifford Algebr., 27 (2), 1801-1813.
- Savin, D. (2019). Special numbers, special quaternions and special symbol elements, chapter appeared in the book Models and Theories in Social System, Springer, 179, 417–430.

## Related papers

## Cite this paper

Rayaguru, S. G., Savin, D. & Panda, G. K. (2019). On some Horadam symbol elements. *Notes on Number Theory and Discrete Mathematics*, 25(2), 91-112, DOI: 10.7546/nntdm.2019.25.2.91-112.