How many prime numbers are there?

You learnt about prime numbers today. Oh..The numbers which have only 2 factors. 1 and the number itself. Remember?

You start wondering how many such numbers are there. You start writing down all of them:

2, 3, 5, 7, 11, 13,…

You still keep going on…

….. 83, 89, 97, 101, 103…..

an on…

….. 503, 509, 521, 523, 541….

You are done discovering 100 prime numbers.

and now wonder….When does this end? What will be the last prime number? What an amazing and prestigious number that would be! The last prime number. What a feeling it would be to be the largest of the set of numbers which are the building blocks of all the numbers!

As you keep going on with your list, you imagine the wonderful day that you would finish discovering all the prime numbers. And then you wonder….’what would I do with all those prime numbers?’ You get an amazing idea. Since prime numbers are the building blocks of all the numbers, you decide you will multiply all these prime numbers together and see what number you get.

You don’t feel like waiting for the day when you finish your list. So, you just assume that you have found all the numbers, and give names to this entire list of ‘yet to be discovered’ numbers.

You assume that there are n such numbers and choose to can call them:

p₁, p₂, p₃, p₄, ……, p_n

and then you multiply all of them together and choose to call the number as N.

N = p₁p₂p₃p₄……p_n

These alphabets don’t tell you much about the number. You try to visualize it by taking some examples, although you know that the list of prime numbers is still in progress.

If there were only 2 prime numbers (2,3), N would be 2 x 3 = 6

If there were only 3 prime numbers (2, 3, 5), N would be 2 x 3 x 5 = 30.

If there were only 4 prime numbers (2, 3, 5, 7), N would be 2 x 3 x 5 x 7 = 210.

You try to think if there is anything this number N can tell you. Is there anything special about the different values of N? Not really! Nothing that you see immediately. Do you?

You go back to your list of prime numbers and observe that the numbers 7, 31 and 211 are present in that list. These numbers are exactly one more than the different values of N. Does that mean that N + 1 is special? Is it always a prime number? Oh wait! If this was true, there would be no limit to the list of prime numbers. Irrespective of how many ever prime numbers you discovered, you could always find a larger prime number N + 1. That would mean that there are infinite prime numbers.

N + 1 = p₁p₂p₃p₄……p_n + 1

You are curios and exited about your new conjecture. You decide to verify it for a few more cases.

If there were only 5 prime numbers (2, 3, 5, 7, 11), N + 1 would be 2 x 3 x 5 x 7 x 11 + 1 = 2311, which also turns out to be a prime number.

If there were only 6 prime numbers (2, 3, 5, 7, 11, 13), N + 1 would be 2 x 3 x 5 x 7 x 11 x 13 + 1 = 30031.

You try to see if 30031 is a prime number, and eventually find the following, which shows that it’s not a prime number.

30031 = 59 x 509 ☹️☹️

The conjecture you made doesn’t hold true, since you found a counter-example. So, N+1 is not always a prime number. So, you take back your claim about infinite prime numbers.

You can’t stop thinking about the number N+1. After all, you were so close to making an amazing discovery. You are still thinking. A thought occurs to you.

If this number N+1 is not a prime number, it should be made up of other prime numbers. Since you had already discovered all the prime numbers (p₁, p₂, p₃, p₄, ……, p_n), it should be made up of some of these prime numbers. Let’s say one such prime number which N + 1 is made up of is p_r, which could be any one of p₁, p₂, p₃, p₄, ……, p_n. So, if p_r is a factor of N+1, dividing N+1 by p_r would leave no remainder.

So, you decide to divide N+1 by p_r to check. What would be the remainder?

(N+1)/p_r = (p₁p₂p₃p₄……p_n + 1)/p_r

(N+1)/p_r = (A multiple of p_r + 1)/p_r

The remainder on the above division (N+1)/p_r would be be 1.

Since the remainder is not zero, p_r is not a factor of N+1.

What does this mean?

So, our assumption that N+1 has one of p₁, p₂, p₃, p₄, ……, p_n as a factor gets contradicted. So, N+1 cannot be made up of any of the prime numbers which you have already discovered.

Irrespective of how many prime numbers you discover, either N+1 will itself be a new prime number or it will be made up of different prime numbers apart from your current list.

N + 1 = p₁p₂p₃p₄……p_n + 1

You are satisfied now. You know for sure that you can go on for eternity and keep adding numbers to your list of primes. There are new primes to be found and there will always be.