65 research outputs found
On Grosswald's conjecture on primitive roots
Grosswald's conjecture is that , the least primitive root modulo ,
satisfies for all . We make progress towards
this conjecture by proving that for all and for all .Comment: 7 page
On consecutive primitive elements in a finite field
For an odd prime power with we prove that there are always three
consecutive primitive elements in the finite field . Indeed,
there are precisely eleven values of for which this is false. For
we present conjectures on the size of such that
guarantees the existence of consecutive primitive elements in
, provided that has characteristic at
least~. Finally, we improve the upper bound on for all .Comment: 10 pages, 2 table
A proof of the conjecture of Cohen and Mullen on sums of primitive roots
We prove that for all , every non-zero element in the finite field
can be written as a linear combination of two primitive roots
of . This resolves a conjecture posed by Cohen and Mullen.Comment: 8 pages; to appear in Mathematics of Computatio
An explicit upper bound for when is quadratic
We consider Dirichlet -functions where is a
non-principal quadratic character to the modulus . We make explicit a result
due to Pintz and Stephens by showing that
for all and for
all .Comment: 17 page
Lehmer numbers and primitive roots modulo a prime
A Lehmer number modulo a prime p is an integer a with
1 ≤ a ≤ p − 1 whose inverse a¯ within the same range has
opposite parity. Lehmer numbers that are also primitive roots
have been discussed by Wang and Wang in an endeavour to
count the number of ways 1 can be expressed as the sum of two
primitive roots that are also Lehmer numbers (an extension
of a question of Golomb). In this paper we give an explicit
estimate for the number of Lehmer primitive roots modulo p
and prove that, for all primes p �= 2, 3, 7, Lehmer primitive
roots exist. We also make explicit the known expression for
the number of Lehmer numbers modulo p and improve the
estimate for the number of solutions to the Golomb–Lehmer
primitive root problem
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