## Frequently Asked Questions

#### What is a prime?

The word prime comes from the Latin, primus, meaning, first. A number is prime if the first time it appears is the start of a times table. For example, 2 is prime because it first appears at the start of the 2 times table. Hence 4, 6, 8, 10, ... cannot be prime; we call non-primes, composite, because they are composed of smaller factors. Similarly 9 is not prime because it appeared earlier in the 3 times table.

To generate a list of primes we work through the set of natural (counting) numbers and if the number has not been encountered in an earlier times table, then it is prime. Consequently 1 is not a prime number. If it were prime then no other number could be prime, as every other number features in the 1 times table.

The primes below one hundred are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89, 97

There are many alternative 'definitions' of primes, but it is important to understand that they are really descriptions, not definitions. They are usually a symptom/consequence of the fundamental definition of primes. For example, a useful description of a prime is a natural number with exactly two divisors – no more and no less.

Eratosthenes, a famous Greek mathematician and friend of Archimedes, used a sieve to find primes. We shall demonstrate this by producing a grid 6 units wide and fill it with the natural numbers. Starting at 2 we work our way through each number and mark it as prime, crossing off any numbers in its times table (as they are no longer prime candidates).

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

It can be seen that a grid 6 units wide is very efficient, because after identifying 2 and 3 we can eliminate four columns (all the numbers under 2 and 3 and the 4 and 6 columns). As a result we observe that all prime numbers greater than 3 are either side of a multiple of 6.

 5 6 7 11 12 13 17 18 19 23 24 25 29 30 31 35 36 37 41 42 43 47 48 49

That is, p = 6k±1, where k is a natural number.

However, it is very important to appreciate that although this formula generates every prime, p > 3, not every number it generates is prime; for example, for k = 4, 6 × 4 + 1 = 25, which is clearly not prime.