An Intuitive Proof to The Infinitude of Primes

Introduction

Prime numbers are numbers with only two factors, $latex 1$ and itself. Eleven is prime since its factors are only $latex 1$ and $latex 11$. Numbers that are not prime are called composite. Fifteen is composite because it has more than $latex 2$ factors. The factors of $latex 15$ are $latex 1$, $latex 3$, $latex 5$, and $latex 15$.

Primes from 1 to 100

White squares show primes from 1 to 100 (via mathandmultimedia.com).

As you count farther, you will observe that the number of primes are getting fewer and fewer. There are $latex 168$ primes between $latex 1$ and $latex 1000$, $latex 135$ primes between $latex 1000$ and $latex 2000$, $latex 127$ primes between $latex 2000$ and $latex 3000$, and $latex 120$ primes between $latex 3000$ and $latex 4000$.┬áNow, the question is, “Is there a largest prime?” The equivalent to this question is, “Are the number of primes finite?” In this post, we are going to prove intuitively that the number of primes is infinitely many. Continue reading