  $ $The aim of this thesis is to lower the bound on square-free primitive
roots modulo primes. Let $g^{\Box}(p)$ be the least square-free primitive root
modulo $p$. We have proven the following two theorems.
  Theorem 0.1. $$g^{\Box}(p) < p^{0.88}\quad\text{for all primes }p.$$ Theorem
0.2. $$g^{\Box}(p) < p^{0.63093}\quad\text{for all primes } p <
2.5\times10^{15} \text{ and }p > 9.63\times10^{65}.$$
  Theorem 0.1 shows an improvement in the best known bound while Theorem 0.2
shows for which primes we can prove the theoretical lower bound.
  After some introductory information in Chapter 1, we will start to prove the
above theorems in Chapter 2. We will introduce an indicator function for
primitive roots of primes in {\S}2.1 and together with results from {\S}1.2.1,
{\S}1.2.3 and {\S}1.2.4 we will outline the first step in proving a general
theorem of the above form. The next two stages in the proof will be outlined in
Chapter 3. These two stages require the introduction of the prime sieve. Before
defining the sieve in {\S}3.2 we will introduce the $e-$free integers which
will play an important role in defining the sieve.
  In {\S}3.2 we will obtain results that do not require computation, including
Theorem 0.2. An algorithm is then introduced in {\S}3.3 which is the last stage
of the proof. There we will complete the proof of Theorem 0.1.
