On consecutive primitive elements in a finite field

Stephen D. Cohen, Tomás Oliveira e Silva, Tim Trudgian

Type: Article

Publication Date: 2015-03-31

Citations: 4

DOI: https://doi.org/10.1112/blms/bdv018

Abstract

For q an odd prime power with q > 169 , we prove that there are always three consecutive primitive elements in the finite field F q . Indeed, there are precisely eleven values of q ⩽ 169 for which this is false. For 4 ⩽ n ⩽ 8 , we present conjectures on the size of q 0 ( n ) such that q > q 0 ( n ) guarantees the existence of n consecutive primitive elements in F q , provided that F q has characteristic at least n. Finally, we improve the upper bound on q 0 ( n ) for all n ⩾ 3 .

Locations

Bulletin of the London Mathematical Society - View
arXiv (Cornell University) - View - PDF
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+	Sets of primitive roots	1958	L. Carlitz
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+	The Magma Algebra System I: The User Language	1997	Wieb Bosma John Cannon Catherine Playoust
+	Pairs of primitive roots	1985	Stephen D. Cohen
+	On consecutive quadratic non-residues: a conjecture of Issai Schur	2003	Patrick Hummel
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+	Distribution of residues and primitive roots	2013	Jagmohan Tanti R. Thangadurai
+ PDF Chat	Pairs of consecutive primitive roots modulo a prime	1968	Emanuel Vegh
+ PDF Chat	A note on Dirichlet characters	1973	Richard H. Hudson
+	A proof of the conjecture of Cohen and Mullen on sums of primitive roots	2014	Stephen D. Cohen Tomás Oliveira E Silva Tim Trudgian