09 Primes
Mathematics is such a massive subject, criss-crossing all avenues of human enterprise,
that at times it can appear overwhelming. Occasionally we have to go back to basics.
This invariably means a return to the counting numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12,... Can we get more basic than this?
Well, 4 = 2 × 2 and so we can break it down into primary components. Can
we break up any other numbers? Indeed, here are some more: 6 = 2 × 3, 8 = 2
× 2 × 2, 9 = 3 × 3, 10 = 2 × 5, 12 = 2 × 2 × 3. These are composite numbers
for they are built up from the very basic ones 2, 3, 5, 7,... The ‘unbreakable
numbers’ are the numbers 2, 3, 5, 7, 11, 13,... These are the prime numbers,
or simply primes. A prime is a number which is only divisible by 1 and itself. You
might wonder then if 1 itself is a prime number. According to this definition it
should be, and indeed many prominent mathematicians in the past have treated
1 as a prime, but modern mathematicians start their primes with 2. This enables
theorems to be elegantly stated. For us, too, the number 2 is the first prime.
For the first few counting numbers, we can underline the primes: 1, 2 , 3 , 4,
5 , 6, 7 , 8, 9, 10, 11 , 12, 13 , 14, 15, 16, 17 , 18, 19 , 20, 21, 22, 23 ,...
Studying prime numbers takes us back to the very basics of the basics. Prime
numbers are important because they are the ‘atoms’ of mathematics. Like the
basic chemical elements from which all other chemical compounds are derived,
prime numbers can be built up to make mathematical compounds.
The mathematical result which consolidates all this has the grand name of the
‘prime-number decomposition theorem’. This says that every whole number
greater than 1 can be written by multiplying prime numbers in exactly one way.
We saw that 12 = 2 × 2 × 3 and there is no other way of doing it with prime
components. This is often written in the power notation: 12 = 2^2 × 3. As
another example, 6,545,448 can be written, 2^3 × 3^5 × 7 × 13 × 37.