## The Prime Numbers

What are prime numbers, and how did they move from the backwaters of abstract mathematical analysis to being vital for the way we live in the computer age? These are a couple of the questions that I look at in this episode of Restful Maths before reading a few out...ok before reading about 500 of them out.

Transcription:

The prime numbers form an infinite subset of the natural numbers; that is to say that they are all positive integers. It is the Greek mathematician Euclid who is credited with the proof that there are infinitely many prime numbers, back in the day, over 2000 years ago. Now although there are infinitely many of them, as we go along the number line they tend to get more difficult to identify, and currently the largest known prime is only about 23 and a quarter million digits long.

A common explanation of what a prime number is is that it can only be divided by itself or one, but this does not stand up as a proper definition in itself. A more elegant definition of what a prime is is that it is a natural number with precisely 2 distinct factors; the only divisible by one and itself idea does not make explicit the distinct nature of the factors, so when asked to list the prime numbers people armed with this concept often start with the number 1, as it cannot be denied that it is divisible by one and itself. But one is not a prime number since it only has one factor.

Throughout the history of classical and modern mathematics from Euclid in 300 BC, through the medieval Islamic mathematicians, the enlightenment and on to today, one has largely been excluded from the primes. Indeed up until the renaissance, whether one was even a number at all for the purpose of analysis was a big can of worms in the world of mathematics.

You notice I say that one has largely been excluded from the list of primes. In the 18th and 19th centuries there were some mathematicians who posited that one is prime, but it was never generally accepted.  Apart from not conforming to our modern definition of what a prime is, acceptance of the primality of one would require a lot of established maths, such as the fundamental theorem of arithmetic, to be redefined to exclude it.

So, what is the first prime number? Well it is also the only even prime number, and that number is 2. Interestingly to some classical Greek mathematicians 2 was not prime as they considered primes to be a subset of the odd naturals. Euclid considered it to be prime, so I guess the ancient Greeks weren’t all barking mad.

With our reliance on computers and the internet the primes have been flung from the backwaters of abstract mathematical analysis and philosophy to one of the cornerstones of how we live today. Computer encryption is dependent on using  prime factors pairs of very large numbers – numbers so large that even a computer is unable to work out which 2 prime numbers were used to make them.