What's the biggest prime number that we know of?

04 September 2017

Question

We hear from time to time they’ve discovered a new highest known prime number, but this is now over 17 million digits. So I know we can't possibly know all the prime numbers up to that. So, what I’d like to know is how far up do we actually know all the primes?

Answer

Chris Smith put this question to maths communicator James Grime...

James - So, a prime number, you may remember from school are those numbers that can only be divided by one and itself. You're right. you will hear in the news announcements about the newest largest prime number we found. The reason that’s important is we do use large prime numbers in encryption and in internet encryption. But also, it’s an exercise in our computers, in computational power. And so, that’s why we have holes. We haven’t been working out the primes one at a time which we can do with ancient methods, by sieving through the numbers, but what we have instead are probabilistic methods to work out extremely high prime numbers. No, we haven’t been working through them one at a time. Where does it stop? Where is that first gap? It must be huge. I do not know that number off the top of my head, but it must be absolute massive.

Chris - Riemann who was a famous mathematician had a prize of I think it’s a million dollars that’s been put up for someone who can come up with a formula for predicting prime numbers but it’s never been claimed.

James - That’s right. that’s the Riemann hypothesis. This must be around 1850. He came up with this function that could be used to work out not where the prime numbers are but where we’re counting the prime numbers. So, we want to know how many prime numbers are less than 100. So, I don’t know what that is. I think it’s about 22 or there must have been. You might want to know how many prime numbers less than a billion. So, we need to know that proportion. Riemann had a method for working out what the error might be in that formula and that will help us predict the pattern in the primes.

Chris - Let’s hope so and then you can make yourself a million, couldn’t you, if you could…

James - And then you can claim…

Chris - You can claim the prize.

James - …this million-dollar prize, absolutely.

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