Introduction

Prime numbers are the building blocks to any conceivable number. Primes are key factors in many mathematical concepts- determining the greatest common denominator and factor, modular arithmetic, the idea of "relatively prime", solving Diophantine equations, and so much more. Prime numbers also have a surprising amount of real-world applications. Primes are used in coloring pixels on the screens of today's technology. Cicadas time their life cycles by primes, emerging once every 13 or 17 years, leading to an unsolved debate as to whether that's simply coincidence or an evolutionary tactic (Webb, 2001). Perhaps the most interesting use for primes is their role in cryptography, allowing for modern computer safety as well as sending secret coded messages. With how ubiquitous they are and still how little is known about prime numbers, primes are considered the most mysterious figures in mathematics.

History and Background

Prime numbers have been a source of intrigue for poeople since the very beginning. It's been known since the times of Euclid that there are an infinite number of primes. The first proof of this appears in the 9th installment of Euclid's "Elements". However, it wasn't known how to easily determine if larger numbers suspected to be primes were, in fact, primes. For centuries, mathematicians labored to find a universal formula for prime numbers, albeit unsuccessfully (Bateman & Diamond, 1996).

Primes also were tangled up in other areas of number theory. One notion that was puzzled and pored over by the Greeks was that of "perfect numbers". Perfect numbers are numbers whose factors add up to the number. For example, the first perfect number is 6, whose factors are 1, 2, and 3, and $1+2+3=6$. Somehow, Euclid managed to detemine a formula for perfect numbers: $2^n-1(2^n-1)$ always yields a perfect number when $2^n-1$ is a prime. Thus, primes were an integral part of the hunt for perfect numbers. Thus, primes were an integral part of the hunt for perfect numbers, and provided valuable information on the mysterious, widely sought-after perfect number. The case of the odd perfect number is the oldest unsolved math problem, dating back to the Greeks, with mathematicians still working on it today. Currently, there are around 51 known perfect numbers. The most recent perfect number was discovered in 2018, having 49,724,095 digits.