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).
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.
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.