Infinite Primes From Integer Partitions

Abstract

Ferrers diagrams are used to visually represent integer partitions. We describe a way to use Ferrers diagrams to uniquely represent integers in terms of their prime factors. This leads to a lower bound on the number of primes less than a given integer, namely $\pi(x) \geq \frac{\lfloor \lg x \rfloor}{\lg (\lfloor \lg x \rfloor + 1)}$ where $\pi(x)$ is the prime counting function and $\lg(x)$ denotes the base 2 logarithm. This results in a new proof of the infinitude of primes.