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ELEMENTS OF LARGE ORDER IN PRIME FINITE FIELDS
Published online by Cambridge University Press: 16 October 2012
Abstract
Given $f(x,y)\in \mathbb Z[x,y]$ with no common components with
$x^a-y^b$ and
$x^ay^b-1$, we prove that for
$p$ sufficiently large, with
$C(f)$ exceptions, the solutions
$(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$ of
$f(x,y)=0$ satisfy
$ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$ where
$c$ is a constant and
${\rm ord}(r)$ is the order of
$r$ in the multiplicative group
$\overline {\mathbb F}_p^*$. Moreover, for most
$p\lt N$,
$N$ being a large number, we prove that, with
$C(f)$ exceptions,
${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$ where
$\epsilon (p)$ is an arbitrary function tending to
$0$ when
$p$ goes to
$\infty $.
- Type
- Research Article
- Information
- Copyright
- Copyright © 2012 Australian Mathematical Publishing Association Inc.
References
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