Prime Numbers Explained a Different Way
Let's consider the positive integers greater than 1, that is 2,3,4,5,..
Suppose we are given the first integer and asked to make all other integers using only the multiply operation. So our set of given integers is g={2}
We soon run into trouble because 2*2=4 and we have no way to make 3.
OK, so we just add 3 to our set of given numbers, so now g={2,3}
Can we make 4? Yes, 2*2=4
Can we make 5? No, all our tries fail, so we add 5 to our set of given numbers g={2,3,5}
Can we make 6? Yes, 2*3=6
Can we make 7? No, all our tries fail, so we add 7 to our given numbers g={2,3,5,7}
Can we make 8? Yes, 2*2*2=8
Can we make 9? Yes, 3*3=9
Can we make 10? Yes, 2*5=10
Can we make 11? No, so we add it to the set g={2,3,5,7,11}
Can we make 12? Yes, 2*2*3=12
Can we make 13? No, so add it to the set g={2,3,5,7,11,13}
Can we make 14? Yes, 2*7=14
Can we make 15? Yes, 3*5=15
Can we make 16? Yes, 2*2*2*2=16
Can we make 17? No, all our tries fail, so we add 17 to our given numbers g={2,3,5,7,11,13,17}
What is the set g that we are generating by this process? It's the set of prime numbers! This is simply another way to explain prime numbers.
It's a nice demonstration because it shows how prime numbers generate all numbers using only the multiply operation. You can also see that as g gets bigger we can obviously make more numbers from it, so prime numbers become less and less frequent.
Content written and posted by Ken Abbott abbottsystems@gmail.com