# The Least Square-free Primitive Root Modulo a Prime

## TL;DR

This thesis improves bounds on the smallest square-free primitive root modulo a prime, providing new theoretical limits and computational results for primes within specific ranges.

## Contribution

It introduces improved bounds on the least square-free primitive root modulo primes and develops a new sieve-based method for proving these bounds.

## Key findings

- Proved that the least square-free primitive root is less than p^{0.88} for all primes p.
- Established that the least square-free primitive root is less than p^{0.63093} for primes p < 2.5×10^{15} and p > 9.63×10^{65}.
- Developed an algorithm to computationally verify bounds for specific prime ranges.

## Abstract

$ $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.