How Many Primes are There
Is Infinity Real
Teachers: The following is a discussion that can be had with students to create interest in mathematics by discussing two very easy to understand, but perplexing problems in mathematics. First, the nature of infinity. The second is the lack of pattern and order in the prime numbers.
The number of primes is infinite. Euclid proved it in a beautiful, easily understood proof by contradiction. Paraphrasing, he said that there are either infinitely many primes, or a finite number of primes. So let’s pick one and explore it. Say there are a finite number of prime numbers. If you were to list them all, then take their product you would have a very large number. But if you just add one to that number, it would be prime because none of the other prime numbers would be a factor of it. It would have exactly two factors, one and itself.
In case you don’t believe this works, let’s say we can list all of the primes, but there are only four. Let’s say the entire list of primes was 2, 3, 5, and 7. Their product, 2 × 3 × 5 × 7 = 210. This number is composite because all of the primes are factors of it. Add one to it, arriving and 211 and none of the prime numbers are a factor of it…making it have the factors of 211 and 1. That means it is prime.
So it is false that there are a finite number of primes. Therefore, the are infinitely many prime numbers.
Beautiful, right? Case closed. … or is it?
The case is closed, if you believe infinity exists. To be clear, infinity is not a number, it’s a concept. A set can only approach infinity, nothing ever equals infinity because it’s an idea. The idea behind infinity is that the collection of things just keeps growing and growing.
We, as humans, have a very big problem with very big numbers, even large groups of things. For example, there are some things that we only have a plural word for, we do not possess a singular word for these things. A few examples are rice, sand, hair, shrimp and fish. You can have a single hair, a grain of sand (or rice), and so on. They are so vast in quantity they become indistinguishable.
And yet, they’re finite. You could conceivably collect all of the sand in the world and count every grain. More sand does not magically appear once it is all collected.
What about stars in the sky? What we call the observable universe is how far we can see. We don’t know if it goes on forever, or if it is somehow contained. Perhaps the word, universe, is misleading. Perhaps there are multiples of it, maybe as many as there are grains of sand on the earth.
Before we chase that rabbit down its hole, let’s get back to earth. Euclid’s proof that there are infinitely many prime numbers is beautiful. But is he right? Surely his proof is flawless, but what about infinity. We have no examples of infinity, it might just be a human construction. Now, if mathematics can discover things that are real and applicable from such a thing, that’s all the more powerful the tool it is, but what if we’re wrong about infinity? There are two things I want you to consider as we explore prime numbers and their relationship with infinity.
The first thing is: There’s an axiom (a statement we just accept as truth), called the Axiom of Infinity. It basically says that there are infinite sets of things, like natural numbers. We just say it’s true and roll with it until we discover a problem. Then, we either adjust our axiom or start a new one.
The second thing is: In the early 20th century a man named Kurt Gödel showed that we cannot actually prove any system of mathematics is true without assuming some supporting evidence is true. We have to assume something is true in order to know if other things are true, roughly speaking. In order to know if the thing we assumed to be true is actually true or not (like infinity), we have to assume that something else, more basic, is true. So, and I’m taking some liberties here to make my point, but a conclusion, like the number of primes being infinite, is only as worthy as the presupposition (infinities exist).
Let’s look at a few strings of prime numbers and see if we can’t get our heads around this whole infinity thing.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
The gaps between these prime numbers are below.
1, 2, 2, 4, 2, 4, 2, 4, 6
Another string would be:
907, 911, 919, 929, 937, 941, 947
The gaps here are listed below.
4, 8, 10, 8, 4, 6
They are still relatively close. Many mathematicians have tried to find a pattern in prime numbers. After all, if you can find a pattern, then you can find the next one. How cool would that be, right?
You might be thinking, uh, why would that be cool?
Well, there’s big money being paid if you can find the next prime number. There is a project called GIMPS (Great Internet Mersenne Prime Search), where you can participate in the search. And if your computer finds the next prime, you get some cash!
The last prime found with GIMPS was in 2013. (At the time of this being written, it is 2017.) The number is massive. The text file of the digits in the number is 7.7 MB. That’s more data that a song and this is just a list of numbers. The number is 257,885,161 – 1. The number is huge that to verify that it is prime takes massive super computers days upon days to perform the calculation. Finding the next prime number is a huge undertaking, very complicated and difficult, requiring computers all over the world working together before one is discovered.
Why all the fuss? What good are they?
Well, they keep you from being robbed, for one. Internet security uses prime numbers to encrypt (code) your banking information. The merchant will have a huge number that they multiply your card number by (kind of). The huge number is the product of two of these gigantic prime numbers. It’s so big that even though everybody (would be thieves) know it’s the product of two primes, they can’t figure out which two numbers. The encrypted number is sent to your financial institution, who knows which two primes were used, which is basically like a key.
It’s also weird, and cool, that some bugs have a life cycle that only occurs in prime numbers! Cicadas only come out and breed, and then die, in prime number years. Incredible.
Back on track, forgive me. It feels there are infinitely many tangents I can follow with math! We have not been able to find a pattern in the prime numbers yet and let’s take a look at why. You see, as these primes get huge, the gaps get larger and larger…approaching infinity!
Let’s take a look at one more string of prime numbers.
10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099
The differences here are as follows.
28, 2, 22, 6, 10, 12, 2, 6
No discernable pattern, right? If you can find one, you stand to make significant history, no one has found one yet. We have some approximations that work within certain constraints but they all break down eventually.
But, to be clear, if you could find a pattern in the gaps between the primes a formula could be created that would generate prime numbers. We can generate natural numbers by just adding one to the largest we have come up with so far. But primes, as you’ve seen with the GIMPS project, aren’t so easily discovered.
And here’s one of the issues. The gaps between prime numbers can get huge, perhaps infinitely huge. Consider this.
Fact 1: 5! = 5×4×3×2 = 120
Fact 2: 120 is not prime because it is divisible by 5 and 4 and 3 and 2.
Fact 3: 5! + 5 is not prime because it is divisible by 5. (When we add another 5, it’s like skip counting when you first learned multiplication.)
The same is true for 5! + 4 being divisible by 4, because 120/4 = 30. 5! + 4 is 4 × 31, there’s one more four.
The same holds true for 5! + 3 being divisible by 3 and 5! + 2 being divisible by 2.
Fact 4: What all this means is that there after 5! + 1 there are four consecutive numbers that are composite.
This would also work for 100! The number 100! + 100 would be composite. For that matter, 100! + 37 would be composite also. 100! Plus all of the numbers up to and including 100 would be composite, (except possibly adding 1).
This means there is a gap of 99 after 100! + 1.
This goes on forever, arbitrarily large numbers, like 1,000,000,000,000! There would be a gap of 1,000,000,000,000 – 1 numbers after this number that are composite.
We could write this in a general sense. Let a and x be a whole numbers such that a is less than or equal to x. (a ≤ x).
Then x! + a is composite.
Since x is a whole number and whole numbers are infinite, then there are infinitely large gaps between the large prime numbers, themselves being infinite.
So if the gaps between primes gets infinitely large, how can there be infinitely many prime numbers?
Well, there’s one more piece of information to be considered. Twin primes are prime numbers that are just two numbers apart. The primes 2 and 3 are only one apart, but all others are an even number apart, the smallest gap being a gap of two, like 5 and 7, or 11 and 13.
There’s a conjecture (not as strong as an axiom), that is yet unproven, but we’re getting closer, that states that there are an infinite number of twin primes. The largest known pair of twin primes is below:
3,756,801,695,685 × 2666,689 – 1
3,756,801,695,685 × 2666,689 +1
Those numbers are too large to be written out!
While we do not yet know, with a proof, that there are infinitely many twin primes, we do know that there are infinitely many primes that have a maximum distance between them and it might be as low as a difference of sixteen. This is all being discovered and explored and fought over at the moment.
So on one hand we have infinitely large gaps between prime numbers, but when they do pop up, they will do so in clumps and groups?
If all of this makes your head spin, then I have succeeded. I am not trying to convince you that infinities do not exist, or that they do. I am trying to show that math is contentious and changing. As we learn and discover new things math is changing. Math is just a language we use to describe the world around us. So powerful is math that we are not even sure if it is a human invention at all or rather a discovery!
What are your thoughts? Please share them in the comments below.
As always, thank you for your time. I hope this has stirred some thought, maybe even sparked a passion for mathematics!
At the time of the making of this video the world’s largest prime number is not the last one found by the GIMPS project. However, they’re likely to find another even larger one, sometime soon. There’s a video below (Largest prime number) that discusses that number and prints it out … it takes up as much paper as three large books!
For some fascinating and approachable treatment of prime numbers, consider the following videos:
Gaps between prime numbers: https://youtu.be/vkMXdShDdtY
The largest prime number: https://www.youtube.com/watch?v=lEvXcTYqtKU
Infinite Primes: https://www.youtube.com/watch?v=ctC33JAV4FI
Large Gaps Between Primes: https://www.youtube.com/watch?v=BH1GMGDYndo
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