Primes – How high can we go?

When we last left off I had finished talking partially about prime numbers and how to find them. But some of you may have been wondering, how many primes there are in total? Is there some way to maybe find a formula to find a prime number?

If you look at the primes, they seem to go on forever! And thanks to amazing mathematicians from the past we know that they do! But HOW do we know that they actually do? How can you be sure there is a number after 97 that is prime? Some will just point to their computer and say, “Computers can prove that!” and they would be partially right. Computers have helped us find extraordinarily HUGE prime numbers. For example, remember those special prime numbers called Mersenne primes which are primes of the form $$2^{p} – 1$$ where p is also prime? Well, a worldwide collaboration was started in order to find the largest prime known to man and they called themselves GIMPS.

The organization is called GIMPS or Great Internet Mersenne Prime Search. You can find their website here: [GIMPS] Quite recently they found the largest prime ever known to man. What is the prime number you may be wondering? It is: $$2^{57,885,161}-1$$. You might be thinking that this looks like a small number, but it really isn’t. This prime is 17,425,170 digits long. That is HUGE! It is so big that if you were to print out the number on paper (without commas) it would be 4,283 pages long!

It took years for the computers to calculate that prime number, so as we questioned earlier, how do we know if there is a prime number greater than this or not? It is so large we can barely use computers to go any higher, and we can forget about writing the number down (you try writing out 4,283 pages of numbers!). So then how can we KNOW if there are infinite number of primes or not? Well this is exactly what a guy named Euclid was pondering over 2,000 years ago.  Euclid was able to prove that there are an infinite number of primes! So no matter how big of a prime number we find, there will always be a bigger one. So cool! But how did Euclid come up with this?

Below is a rough transcript of how Euclid created his proof, but before we go over the proof here a lemma that Euclid uses in order to prove it that you should know:
Lemma 1: Every number greater than 1 can be broken down into multiples of prime numbers. e.g: $$12=2 \cdot 2 \cdot 3, 4=2 \cdot 2, 3=3, 5=5$$. Notice that prime numbers just equal themselves, and composite numbers are multiples of primes!

Euclid’s Proof: Suppose that there are only a finite number of primes. This means that there is a number x such that any number greater than x is not prime. All these numbers greater than x thus must be composite and by the lemma we just discussed must be a combination of prime numbers. So if we take every prime number less than x and multiply them together we get a number n. Now if we add 1 to n we get a prime number! (Notice that since 2 is a prime, thus multiplying all the primes together would get us an even number, which would automatically not be prime. So we add 1 to make it an odd number.) Why/how is n+1 prime?! Because no matter which prime number we select and we divide n + 1 by it we will get a remainder of 1. And since we have used every prime number less than x, and every number between x and n are composite, there are no other prime numbers that (n+1) can divide into. So (n+1) must be prime! But, we had stated earlier that x is the greatest prime and since (n+1)>x we see that our earlier thought was wrong (that there are a finite number of primes) and so there must be an infinite number of primes!

[Note: This type of proof is called proof by contradiction. I’ll eventually lay the groundwork for how this works, but for now you can look it up in google if you’d like.]

Wow, that was complex, but so rewarding! We can now for sure state that there are an infinite number of primes! Fun part is, that this was only 1 of many proofs that there are infinite primes! Any questions? Leave a comment!

Questionably yours,
The Cali Garmo