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 | References | Similar Articles | Additional Information

Abstract: Let be an odd prime and . Let be an odd positive integer. Let or and . By employing the integer sequence , which can be considered as a generalized Lucas sequence, we construct all the permutation binomials of the finite field .

**1.**Leonard Eugene Dickson,*The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group*, Ann. of Math.**11**(1896/97), no. 1-6, 161–183. MR**1502221**, 10.2307/1967224**2.**Michael D. Fried, Robert Guralnick, and Jan Saxl,*Schur covers and Carlitz’s conjecture*, Israel J. Math.**82**(1993), no. 1-3, 157–225. MR**1239049**, 10.1007/BF02808112**3.**D. R. Hayes,*A geometric approach to permutation polynomials over a finite field*, Duke Math. J.**34**(1967), 293–305. MR**0209266****4.**C. Hermite, Sur les fonctions de sept lettres,*C. R. Acad. Sci. Paris***57**(1863), 750-757;*Oeuvres*, vol. 2, pp. 280-288, Gauthier-Villars, Paris, 1908.**5.**Rudolf Lidl and Harald Niederreiter,*Finite fields*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. MR**1429394****6.**M. O. Rayes, V. Trevisan and P. Wang, Factorization of Chebyshev polynomials,*http://icm.mcs.kent.edu/reports/index1998.html*.**7.**Theodore J. Rivlin,*The Chebyshev polynomials*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0450850****8.**N. J. A. Sloane,*The On-Line Encyclopedia of Integer Sequences*, Published electronically at http://www.research.att.com/njas/sequences/.**9.**Da Qing Wan and Rudolf Lidl,*Permutation polynomials of the form 𝑥^{𝑟}𝑓(𝑥^{(𝑞-1)/𝑑}) and their group structure*, Monatsh. Math.**112**(1991), no. 2, 149–163. MR**1126814**, 10.1007/BF01525801

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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

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

DOI:
http://dx.doi.org/10.1090/S0002-9939-05-08220-1

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.