Four consecutive primitive elementsin a finite field

Jarso, Tamiru; Trudgian, Tim

doi:10.1090/mcom/3716

Four consecutive primitive elementsin a finite field
HTML articles powered by AMS MathViewer

by Tamiru Jarso and Tim Trudgian HTML | PDF

Math. Comp. 91 (2022), 1521-1532 Request permission

Abstract:

For $q$ an odd prime power, we prove that there are always four consecutive primitive elements in the finite field $\mathbb {F}_{q}$ when $q> 2401$.

References

W. Bosma, J. Cannon, C. Fieker, and A. Steel, The Magma algebra system. I. Version 2.25, January 21, 2020. http://magma.maths.usyd.edu.au/magma/.
A. Brauer, Über sequenzen von potenzresten, S.-B. Preuss. Akad. Wiss., 1928, 9–16.
L. Carlitz, Sets of primitive roots, Compositio Math. 13 (1956), 65–70. MR 83002
Stephen D. Cohen, Consecutive primitive roots in a finite field, Proc. Amer. Math. Soc. 93 (1985), no. 2, 189–197. MR 770516, DOI 10.1090/S0002-9939-1985-0770516-9
Stephen D. Cohen, Consecutive primitive roots in a finite field. II, Proc. Amer. Math. Soc. 94 (1985), no. 4, 605–611. MR 792270, DOI 10.1090/S0002-9939-1985-0792270-7
Stephen D. Cohen, Pairs of primitive roots, Mathematika 32 (1985), no. 2, 276–285 (1986). MR 834496, DOI 10.1112/S0025579300011050
Stephen D. Cohen, Tomás Oliveira e Silva, and Tim Trudgian, On consecutive primitive elements in a finite field, Bull. Lond. Math. Soc. 47 (2015), no. 3, 418–426. MR 3354437, DOI 10.1112/blms/bdv018
Richard H. Hudson, A note on Dirichlet characters, Math. Comp. 27 (1973), 973–975. MR 337850, DOI 10.1090/S0025-5718-1973-0337850-8
Patrick Hummel, On consecutive quadratic non-residues: a conjecture of Issai Schur, J. Number Theory 103 (2003), no. 2, 257–266. MR 2020271, DOI 10.1016/j.jnt.2003.06.003
M. Hunter, The Least Square-free Primitive Root Modulo a Prime, Honours Thesis, ANU, 2016.
Tamiru Jarso and Tim Trudgian, Quadratic non-residues that are not primitive roots, Math. Comp. 88 (2019), no. 317, 1251–1260. MR 3904145, DOI 10.1090/mcom/3378
Kevin McGown, Enrique Treviño, and Tim Trudgian, Resolving Grosswald’s conjecture on GRH, Funct. Approx. Comment. Math. 55 (2016), no. 2, 215–225. MR 3584569, DOI 10.7169/facm/2016.55.2.5
J. M. Pollard, Theorems on factorization and primality testing, Proc. Cambridge Philos. Soc. 76 (1974), 521–528. MR 354514, DOI 10.1017/s0305004100049252
Emanuel Vegh, Pairs of consecutive primitive roots modulo a prime, Proc. Amer. Math. Soc. 19 (1968), 1169–1170. MR 230680, DOI 10.1090/S0002-9939-1968-0230680-7