## 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` = 6`k`±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.