## How to find a giant prime number

Recently, mathematicians used computers to find the largest prime yet. It’s really big – it has over 17 million digits. If you printed it out in books, it would be roughly three times longer than all the Harry Potter books combined.

To make the number simpler to communicate, mathematicians have a different way of writing it: 257 885 161 – 1. This expression gives you a way of calculating the new prime number. First, take 57 885 161 ‘2’s and multiply them all together. Then subtract 1.

Numbers that can be written like this – by multiplying 2s and then subtracting 1 – are called Mersenne numbers. The first few Mersenne numbers are 1, 3, 7, 15, 31, 63, 127 and 255. Not all Mersenne numbers are primes and not all primes are Mersenne numbers. But Mersenne primes have some really useful patterns that help when looking for large primes.

If you look at the sequence closely, you’ll notice that every second Mersenne number is divisible by three. This means you don’t have to check those numbers – they can’t be primes. There are similar patterns in the sequence, too. For example, every third term is divisible by 7. In fact, you only have to check Mersenne numbers where you multiply a prime number of 2s together – the rest can’t be prime.

This trick doesn’t do all the work of finding large primes. Not all these remaining Mersenne numbers are prime, so they need to be checked. But all this clever thinking helps mathematicians to take a good guess at which Mersenne numbers might be prime. This is important, because it takes a lot of computer power to check them. It took 39 days of non-stop computing to find that 257 885 161 – 1 was prime. And it’s only the 48th Mersenne prime found!

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