Largest Prime Now Has Over 23 Million Digits
Written by Mike James
Sunday, 07 January 2018

The newly discovered prime is 277,232,917-1. There is no real practical use for a prime of this size, it is simply for the fun and the mathematics of it. And I suppose the technical accomplishment of having the computing power to verify that it is a prime.

Primes are very simple to define - they are numbers that have no factors, i.e. you can't find a smaller number that divides into them without a remainder. All non-primes can be written as the product of nothing but primes. This makes primes something like the atoms of numbers from which every other number can be derived.

The first surprise is that there are a lot of primes - an infinity to be precise - and they aren't even rare in any given range. If you go looking for primes you are sure to fall over a few thousand without even trying hard.

So what is the fuss about finding a new prime?

There are two answers. The first is that it is the largest prime of a particularly interesting type and secondly it is the largest prime of any type we know of.

The number in question is  277,232,917– 1 and it has more than 23 million digits. It is called M77232917 for obvious reasons and it was proved to be prime on December 26th 2017.

Marin Mersenne 1588-1648

It is an example of a Mersenne prime which are all of the form 2p-1 but not all numbers of this form are prime. In particular if p isn't prime then 2p-1 isn't prime. For example, 22-1 or 3 is a prime and so is 23-1=7 is also but 24-1 =15 isn't since 4 isn't a prime.

The sequence for p that give primes is 2,3,5,7,13,17,19,31...

They become fewer as we go on and with the addition of M77232917 we now have just 50 Mersenne primes. We don't even know if the Mersenne primes go on forever or stop - which means M77232917 could be the last of the Mersenne primes.

The Great Internet Mersenne Prime Search, more commonly referred to as GIMPS, has organized the search since 1996 and has found the last 16 Mersenne primes. You can download a program to help you search for the next prime and there is even a small reward if you are the lucky one.

The press release about the recent find reports:

"Jonathan Pace is a 51-year old Electrical Engineer living in Germantown, Tennessee. Perseverance has finally paid off for Jon - he has been hunting for big primes with GIMPS for over 14 years. The discovery is eligible for a \$3,000 GIMPS research discovery award."

But be warned, he almost certainly used more electricity over the 14 years than the prize money could pay for.

"The primality proof took six days of non-stop computing on a PC with an Intel i5-6600 CPU. To prove there were no errors in the prime discovery process, the new prime was independently verified using four different programs on four different hardware configurations."

Why compute such huge primes?

The only reasonable answer is the mountain-climber riposte: because it's there.

There are no practical uses for Mersenne primes or primes of such size. They are too big for cryptography and most of the work in number theory about them doesn't require actual examples of numeric values. Perhaps one day someone will find a use but somehow I doubt it - but I'd be happy to be wrong.

UPDATE: Another Mersenne Primes has been discovered see: A New Mersenne Prime Discovery (51st)

GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1

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