As a reminder, a product is the result of multiplication of factors. So if we multiply the factors a, b, c and d, we obtain the product abcd.
We can explore whole numbers (it is enough considering the positive ones) and see whether they can be expressed as products of factors. For example, 4 = 2·2 and 8 = 2·2·2.
The number 1 must not be considered a factor. Otherwise we can express any number as an arbitrarily crazy multiplication of factors, like 3 = 3·1 = 1·1·3·1 = 1·3·1·1·1·1, etc. There is no point in that.
Once 1 is excluded as a factor, we see how there are special numbers which cannot be factorised (decomposed into smaller factors).
These numbers, from the smallest and in increasing order, are
2, 3, 5, 7, 11, 13, 17, etc.
They are called *simple* or *prime* numbers.
On the other hand, the rest of numbers,
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, etc,
can be decomposed into factors, and they are called *composite* numbers.
Prime numbers form a succession that appears to be irregular. If we observe the increment between two consecutive primes, there is no apparent order.
No one has been able to discover a simple law that predicts these numbers.
All composite numbers can be factorised into prime factors. If we would obtain a non-prime factor, such factor could be further factorised.
Another important aspect of this is that the factor decomposition of each number is *unique*. In fact, the fact that each number has a unique prime factorisation constitutes what we call the *fundamental theorem of arithmetic*. It is also called the unique factorisation theorem.
The number 2 has factorisation 2, and 3 has factorisation 3. The number 4 has factorisation 2·2, and so on. Each factorisation is the unique signature of a number. A number like 120 can be uniquely characterised by the factorisation 2·2·2·3·5.
If we observe the factorisation signatures of primes we see that some have a single factor: these are the primes. Others have several factors: these are the composites. But within the composites, there are numbers which exhibit each factor only once, and others in which some factors are repeated.
This is a good place to introduce the two *prime omega* functions: ω and Ω.
The little omega function ω(n) is a function that, given a positive integer n, it returns the amount of *distinct* prime factors.
The big omega function Ω(n), in contrast, returns the amount of prime factors, in total, without regard of whether they are repeated or not.
Let's explore the values of ω and Ω for some numbers:
+------+---------------+---------+---------+ | n | factorisation | ω | Ω | +------+---------------+---------+---------+ | 2 | 2 | 1 | 1 | +------+---------------+---------+---------+ | 3 | 3 | 1 | 1 | +------+---------------+---------+---------+ | 4 | 2·2 | 1 | 2 | +------+---------------+---------+---------+ | 5 | 5 | 1 | 1 | +------+---------------+---------+---------+ | 6 | 2·3 | 2 | 2 | +------+---------------+---------+---------+ | 7 | 7 | 1 | 1 | +------+---------------+---------+---------+ | 8 | 2·2·2 | 1 | 3 | +------+---------------+---------+---------+ | 9 | 3·3 | 1 | 2 | +------+---------------+---------+---------+ | 10 | 2·5 | 2 | 2 | +------+---------------+---------+---------+ | 11 | 11 | 1 | 1 | +------+---------------+---------+---------+ | 12 | 2·2·3 | 2 | 3 | +------+---------------+---------+---------+ | 13 | 13 | 1 | 1 | +------+---------------+---------+---------+ | 14 | 2·7 | 2 | 2 | +------+---------------+---------+---------+ | 15 | 3·5 | 2 | 2 | +------+---------------+---------+---------+
It is clear from the table that prime numbers have ω = Ω = 1.
If we are given a number, like 360, we can explore its factors in an easy way. For example, as the number is even, we can see that 360 = 2·180, so 2 is already a factor. But 180 is even again, as 180 = 2·90. And 90 is even, so that we have the factor 2 three times: 2·2·2. But now we are left with 45, which is odd.
When we reach an odd, we will never obtain an even, so it is time to try higher factors. For example 3. We see that 45 = 3·15, so that 3 is a factor. And 15 is 3·5, which completes our quest: 360 = 2·2·2·3·3·5.
This is the prime factorisation of the number 360, and from it, we can see that ω(360) = 3, and Ω(360) = 6.
The previous method to factorise numbers would benefit from the operation of division, which is the subject of our next chapter.