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$ $The aim of this thesis is to lower the bound on square-free primitiveroots modulo primes. Let $g^{\Box}(p)$ be the least square-free primitive rootmodulo $p$. We have proven the following two theorems.  Theorem 0.1. $$g^{\Box}(p) < p^{0.88}\quad\text{for all primes }p.$$ Theorem0.2. $$g^{\Box}(p) < p^{0.63093}\quad\text{for all primes } p <2.5\times10^{15} \text{ and }p > 9.63\times10^{65}.$$  Theorem 0.1 shows an improvement in the best known bound while Theorem 0.2shows for which primes we can prove the theoretical lower bound.  After some introductory information in Chapter 1, we will start to prove theabove theorems in Chapter 2. We will introduce an indicator function forprimitive 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 generaltheorem of the above form. The next two stages in the proof will be outlined inChapter 3. These two stages require the introduction of the prime sieve. Beforedefining the sieve in {\S}3.2 we will introduce the $e-$free integers whichwill play an important role in defining the sieve.  In {\S}3.2 we will obtain results that do not require computation, includingTheorem 0.2. An algorithm is then introduced in {\S}3.3 which is the last stageof the proof. There we will complete the proof of Theorem 0.1.

The Least Square-free Primitive Root Modulo a Prime