We were given very simple definition by our teachers about what is a prime number and it was “a number not divisible by any other number except number 1 and itself is prime”. It was a cool definition. Let’s think about it bit differently.
Imagine a unicorn which can only jump 2 places and is standing on apple nbr 2. If we let unicorn run on below apple road how many apples will it crush ?
In first jump it will crush apple-4. And it misses apple 3. And further it will crush 6,8,..
Let’s put a rule, for every jump if it cannot crush a apple – apple becomes a unicorn of that number. And that unicorn then jumps with earlier unicorn. But here the jump length of that new unicorn is the new uncrushed apple number.
So we get our first new unicorn nbr 3. Now if both make jump we will see below :
They cannot crush apple nbr 5. So we get another unicorn of number 5.
Now if all of them makes jump, we see they all cannot crush apple 7. All other apple get crushed.
What have we seen so far?
Every unicorn that gets created due to apple not being crushed, is a prime unicorn. 3, 5 ,7 numbered unicorn are all prime unicorn and are the prime numbers below 10.
We only needed unicorn 2 ,3 and 5 to find out if there are any other prime unicorn below 10. (this modifies our initial definition, we don’t need to divide a number by every other number, but instead every other prime number before it (but hold on to it as well for future blog)).
ie. we found that a apple becomes a unicorn if it does not come under the cycles of earlier unicorns.
So we can refine the earlier definition to “A number is a prime if it cannot be divided by primes smaller than it. And 2 is a first prime.”
Important question remains – how we chose our first unicorn. Why was that unicorn numbered 2 and not 1 or 3. My answer is – it is how humans think. We make assumptions to create beautiful problem definition. If we have chosen first unicorn as 1 then, no apple would be left and then no unicorns and no prime numbers.
And now humans are still finding all those big unicorns which are apples and could not be crushed by earlier unicorns. As of Today the largest found prime number is mentioned here.
For ref, below is the further sequence from above concept. It generates unicorns of 11 13 and 17 value.