#### Prime Number Series

2, 3, 5, 7, 11, … . What do these numbers have in common? They are all primes: they are not divisible by any number except 1 and themselves. That is an intriguing property for a number to have. It reminds us of the fundamental particles in physics; indivisible units that make up all the atoms and molecules we find in nature.

And indeed, the primes do play such a fundamental role in mathematics. If you take a whole number that is not prime, say 12, you can break it down into its divisors, for example 12=2×6=3×4. Each divisor that is not prime itself can be further broken down: 12 = 2×6 = 2x2x3 and 12=3×4=3x2x2. Every divisor in this example is now prime, so we stop. For a number other than 12 you may have more steps to go, but ultimately you will also end up with an expression of your number as a product of primes. And what is more, this expression is unique: the primes might appear in a different order but any such expression of a number will always contain exactly the same primes.

This result, that every whole number is a product of primes in a unique way, is called the fundamental theorem of arithmetic. It made its earliest appearance in Euclid’s famous book The Elements in around 300BC. Euclid also showed that there are infinitely many primes: they never run out as you move up the number line.