2,900 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
Triples and quadruples of consecutive squares or non-squares in a finite field
\begin{abstract} Let \F be the finite field of odd prime power order ,
We find explicit expressions for the number of triples \{\al-1,\al,\al+1 \}
of consecutive non-zero squares in \F and similarly for the number of triples
of consecutive non-square elements. A key ingredient is the evaluation of
Jacobsthal sums over general finite fields by Katre and Rajwade. This extends
results of Monzingo(1985) to non-prime fields. Curiously, the same machinery
alows the evaluation of the number of consecutive quadruples \{\al -1,
\al,\al+1, \al +2\} of square and non-squares over \F, when is a power
of 5. \end{abstract
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