# Patterns and primes

Mathematicians have always been fascinated by prime numbers – the raw material from which all numbers are made. A prime number is any number that can only be divided by two different numbers, itself and 1, without producing a remainder. (The number 1 isn’t considered a prime number, since it can only be divided by itself).

Every other number – the ‘non-primes’ – can be produced by multiplying together a combination of two or more prime numbers, and every such combination is unique. As a result, everything that can be counted is actually a combination of one or more prime numbers. Prime numbers are the foundation of the universe.

The problem for mathematicians is that prime numbers are a mystery. Many wonderful things – and many very complicated things – are known about prime numbers. But so far no-one has ever solved the problem of predicting exactly where prime numbers will occur amongst their non–prime colleagues.

Of course, the list of known prime numbers is staggeringly long. Even before the advent of computers, there were huge tables of prime numbers. But every one of the numbers on that list was originally discovered by someone (or something, nowadays) looking at the number and starting the laborious task of deciding whether can be divided by something smaller, without leaving a remainder.

Today, all the prime numbers are known up to unimaginable values. The largest known has no less than 17,425,170 digits – and that’s undoubtedly out of date. But for all the sophistication of modern methods, the only way the next larger one will be found is by looking at successively larger numbers and asking, ‘is there something smaller that will divide this’.

Much of the work that has been done on the nature and frequency of primes is stratospheric in its complexity – we are literally talking about work which would be understood by only a handful of mathematicians in the world. Yet the fascination of primes is that they are there in plain view, just as the skies are visible to both amateur and professional astronomers. And just as amateur astronomers continue to make discoveries which have eluded the professionals, perhaps one day it will be an amateur mathematician, staring in fascination,, who will come up with the next big discovery.

In the designs in this section, I place primes into a grid of some kind – it’s kind of an obsession. Something in the back of my mind says that if I could just find the right kind of grid, a pattern would become clear. You can always try for yourself putting numbers into some kind of pattern and then marking the primes – spiralling out from 1, or following the outline of a star, or a leaf. One day, someone doing just that may look at the resulting pattern and say: ‘So that’s how prime numbers work!’ Their name would never be forgotten.

Primes 1000

This version of the Sieve of Eratosthenes was just for fun. A very simple program using the graphics oriented language, processing, produced this 25*40 grid representing the numbers up to 1000. Each prime up to 31 (there are 12) is represented by a different colour. The limit of 31 is used because when testing for prime numbers, we only have to try and divide by prime numbers up to the square root of the upper limit (in this case 1000). Since 37 (the next prime number after 31) squared is more than 1000, so 37 times anything bigger will also be more than 1000. It follows that every non-prime number up to 1000  that is, a number made up of two or more prime numbers multiplied together  must be divisible by some prime number less than 37.

Past 31 each number is represented by the colour of the highest prime which will divide it without remainder. In the Sieve of Eratosthenes it is the lowest prime dividing the number but this makes a nicer pattern!

Primes in a hexagon

The first 271 numbers are arranged this time around a hexagonal spiral, with just the prime numbers highlighted.