Primitive Roots
===============

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As a  consequence of
:cite:`McGown-Trudgian20`,
we find the next result.

 
.. admonition:: Theorem (2020)
   :class: thm-tme-emt

   The least primitive root :math:`g(p)` modulo the prime :math:`p` satisfies
   :math:`g(p)\le p^{5/8}` when :math:`p\ge 10^{22}` and :math:`g(p) < \sqrt{p}-2` when
   :math:`p\ge 10^{56}`.


From
:cite:`McGown-Trevino-Trudgian16`,
we also have:

 
.. admonition:: Theorem (2016)
   :class: thm-tme-emt

   Under GRH, the least primitive root :math:`g(p)` modulo the prime :math:`p` satisfies
   :math:`g(p)  409`.



Similar investigations concerning primivite roots modulo :math:`p^2` are led
in
:cite:`Kerr-McGown-Trudgian20`
and in
:cite:`Chen22`
where the next theorem is proved.

 
.. admonition:: Theorem (2022)
   :class: thm-tme-emt

   The least primitive root :math:`h(p)` modulo :math:`p^2` satisfies
   :math:`h(p)\le p^{0.74}` for all :math:`p \ge 2`.