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How to Prove There Are Infinite Prime Numbers?

What's the biggest prime number? Is it the 17,425,170 digit prime discovered in January 2013? While that may be the largest prime number known today, it's certainly not the biggest...because there isn't a biggest! Why? Keep on reading to find out.

By
Jason Marshall, PhD
June 7, 2014
Episode #201

Page 1 of 2

Infinite Spiral ClockToday we're going to talk about two things that every math fan loves: prime numbers and infinity. In particular, we're going to talk about how it is that we can be sure that there are an infinite number of prime numbers. Because there are an infinite number of them, right?

Well, it certainly seems like there should be, but how do we know without a doubt that there are indeed an infinite number of prime numbers? In other words, how can we use math to prove it? Stay tuned because that's exactly the question we'll be answering today!

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What Are Prime Numbers?

Before we start proving anything about prime numbers, we'd better first remind ourselves just what prime numbers are all about. The most basic place to start is with the idea that a prime number is any integer that is only divisible by itself and the number 1. Put another way, a prime number is any number that cannot be created by multiplying two smaller numbers together.

Numbers that aren't prime are known as composite numbers. And the amazing thing is that every one of the infinite number of composite numbers can be built by multiplying prime numbers together—that’s why they’re called composite numbers! For example, the composite number 35 can be built by multiplying the primes 5 and 7, the composite number 56 can be built by multiplying the primes 2 x 2 x 2 x 7, and so on.

Which—and I think you'll agree with me that this is a very cool fact—means that we can think of prime numbers as the fundamental building blocks from which all other positive integers are created!

What Is the Largest Prime Number?

But exactly how many of these prime number building blocks are there? Is there a limit to how big they can be? Well, in some sense it feels like prime numbers must go on and on getting bigger and bigger forever. After all, that's what integers do—why shouldn't prime numbers do it, too? And while that seems reasonable, the truth is that seeming reasonable doesn't make it so. In math, we need to prove it.

In math, we need to prove it.

But before we prove that there is no upper-limit to the size of prime numbers, let's take a quick look at the state-of-the-art in the prime number finding game. The history of how people have looked for larger and larger prime numbers is actually rather fascinating. As you can imagine, searching through the infinitely large stack of integers for large prime numbers is a ton of work, and for a long time before the availability of powerful non-human computers, it was painful, slow, and downright laborious work.

In 1951, before the dawn of electronic computer wizardry, the largest known prime number was 44 digits long. Yes, that's a pretty big number, but it's teeny-tiny compared to the 17,425,170 digit prime number that was discovered in January 2013. If progress continues along the current trend, the first 1 billion digit prime will be discovered within a few decades (perhaps even sooner).

But will this trend ever stop? Does the list of primes go on and on forever?

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