If primes are the atoms of arithmetic, can’t we just make a list of all the primes? After all in Chemistry the Periodic Table lists all the 109 possible atoms in Nature. Can’t we just make a Mathematical Periodic Table of primes?

The Greek mathematician Euclid discovered 2000 years ago that making a list of all the primes is impossible. He recorded this discovery in one of the most famous mathematics books of all time: The Elements.

Proposition 20 of Euclid’s Elements stated:

There are infinitely many primes

Gone is the chance then to list 109 primes and be finished. But how could Euclid really be sure that the primes never run out?

This is the power of mathematical proof.