## A generalized Lucas sequence and permutation binomials

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- by Amir Akbary and Qiang Wang
- Proc. Amer. Math. Soc.
**134**(2006), 15-22 - DOI: https://doi.org/10.1090/S0002-9939-05-08220-1
- Published electronically: July 21, 2005
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## Abstract:

Let $p$ be an odd prime and $q=p^m$. Let $l$ be an odd positive integer. Let $p\equiv -1~(\textrm {mod}~l)$ or $p\equiv 1~(\textrm {mod}~l)$ and $l\mid m$. By employing the integer sequence $\displaystyle {a_n=\sum _{t=1}^{\frac {l-1}{2}} {\left (2\cos {\frac {\pi (2t-1)}{l}}\right )}^n}$, which can be considered as a generalized Lucas sequence, we construct all the permutation binomials $P(x)=x^r+x^u$ of the finite field $\mathbb {F}_q$.## References

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## Bibliographic Information

**Amir Akbary**- Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive West, Lethbridge, Alberta, Canada T1K 3M4
- MR Author ID: 650700
- Email: akbary@cs.uleth.ca
**Qiang Wang**- Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
- Email: wang@math.carleton.ca
- Received by editor(s): July 27, 2004
- Published electronically: July 21, 2005
- Additional Notes: The research of both authors was partially supported by NSERC
- Communicated by: Jonathan M. Borwein
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**134**(2006), 15-22 - MSC (2000): Primary 11T06
- DOI: https://doi.org/10.1090/S0002-9939-05-08220-1
- MathSciNet review: 2170538