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A generalized Lucas sequence and permutation binomials

Authors: Amir Akbary and Qiang Wang
Journal: Proc. Amer. Math. Soc. 134 (2006), 15-22
MSC (2000): Primary 11T06
Published electronically: July 21, 2005
MathSciNet review: 2170538
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Abstract: Let $p$ be an odd prime and $q=p^m$. Let $l$ be an odd positive integer. Let $p\equiv -1~({\rm mod}~l)$ or $p\equiv 1~({\rm mod}~l)$ 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 $\mathbb{F} _q$.

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Additional Information

Amir Akbary
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive West, Lethbridge, Alberta, Canada T1K 3M4

Qiang Wang
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6

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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.