To understand what a prime number is, it is first necessary to revisit the idea of a ‘factor’. Take the number ‘6’, for example: since 6 can be divided by the numbers 1, 2, 3 and 6, these numbers are said to be factors of 6. Before we proceed any further, it is worth noting that we are only considering whole numbers or ‘integers’, as they are called in this context.
We are now in a position to define prime numbers: a prime number is a number with exactly two factors. We can see immediately that 6 falls foul of this definition and is therefore not prime. Let us consider some other small integers. The number 5 can only be divided by 1 and 5 and is therefore prime. On the other hand, 4 has three distinct factors 1, 2 and 4 and so is not prime. It is easy to see that 2 and 3 are prime but what about 1? Many people have learnt to define prime numbers as numbers that are divisible by both 1 and themselves. According to this definition, 1 should be prime, but it is not! That is why the definition by number of factors is generally preferred nowadays it excludes 1.
At this point, you may be wondering why it is important that 1 should be excluded from the world of prime numbers. To understand this, you need to understand why prime numbers are important to mathematicians. This is best explained with the help of an analogy: just as DNA gives a unique description of every living thing, so prime numbers give a unique description of every integer. It is the uniqueness that is really important here. The ancient Greeks realised that every integer, with the exception of 1, was either prime or could be expressed as a product of primes. Let us consider the first few integers: 2, 3, 5 and 7 are prime, but what of the others? 4 can be expressed as 2 x 2, 6 as 2 x 3, 8 as 2 x 2 x 2 and 9 as 3 x 3. There are other ways of expressing these numbers, of course, but these other ways will involve numbers that aren’t prime(we call such numbers ‘composites’). For example, we could have written 8 = 2 x 4, but 4 isn’t prime. If we limit ourselves to the primes, then each composite number can be expressed as a product of primes in one way only. Therefore, each whole number has a unique ‘prime DNA’ associated with it.
The study of prime numbers is ancient and in view of its apparent simplicity it may be tempting to think that it is of no importance to the modern mathematician. Nothing could be further from the truth! Not only are prime numbers used for internet encryption but there are many fascinating secrets regarding primes which have not yet been unlocked. The most famous unsolved problem involving prime numbers is probably the Riemann Hypothesis. That is another story!