Numbers that can only be evenly divided by one and themselves?

Well it turns out finding prime numbers is not only useful for protecting data, but it

could make you money!

On December 26th, 2017, the largest prime number ever discovered was found by Jonathan

Pace of Germantown, Tennessee.

The number is over 23 million digits long, meaning I don't really have time in this

video or my entire life to list the whole thing out, but it's easier to call it by

its nickname, M77232917.

It gets this name because that's the exponent you raise 2 by to find it.

Oh, and don't forget to subtract one when you're done multiplying 77,232,917 twos

together.

Otherwise you just created a number you can divide by two and you'll have wasted all

that time.

Prime numbers found this way, by raising two by a prime exponent and subtracting one, are

called Mersenne primes, named for a 17th century French friar, hence the capital “M” in

the shorthand name.

3 is a Mersenne prime, since it's two to the second power minus one.

Same for 7, which is two to the third power minus one.

But the exponent has to be prime.

Two to the sixth power minus one is 63, which is divisible by 3 so, not a prime.

9 of the 10 largest primes found to date are Mersenne primes, not because they're the

most common -- the Prime Number theorem tells us there must be a simply huge number of undiscovered

primes between this newly found one and the next largest Mersenne prime -- but we find

them because that's how we keep searching for them.

In fact you can be a part of the search for the next one!

You don't even have to do the math yourself, just some software provided by the Great Internet

Mersenne Prime Search, or GIMPS!

The software uses the idle processing power of your CPU to run what's called the Lucas-Lehmer

test, where Mersenne numbers are checked against a specific set of numbers.

If the Mersenne number you're checking divides evenly into a certain number in that set,

then it passes the test and is indeed a prime number.

Obviously it takes quite a bit of horsepower to see if a 23 million digit number divides

into an even bigger number, which is why the project needs the public's help.

Almost 200,000 users are running the GIMPS software and if their PC runs a number that

passes the test, they could win money.

Pace is eligible for a $3,000 prize, and whoever finds a 100 million digit prime could win

$150,000 dollars.It's like cryptocurrency mining for those of us that can't afford

to buy a dozen graphics cards.

You may wonder why we bother searching for primes at all?

Why for the glory, of course!

We've only found 50 Mersenne Primes and next to almost every one of them is the finder's

name for all of nerdy eternity.

Plus prime numbers play a crucial role in keeping data secure.

One standard, RSA encryption, relies on multiplying two large prime numbers together to generate

a key.

Knowing the two primes that went into the key is the secret to decrypting the data,

but it takes an impractical amount of computational power and time to suss out what those numbers

are if you don't already know them.

So the key can be public and your data is still secure.

Of course since the Mersenne primes we've been finding lately are tens of millions of

digits long, these may not be the best candidates for encryption.

If you see a key that's an absolutely huge number, it won't be hard to guess which

Mersenne prime went into it.

Gigantic primes won't be useful until quantum computing is used to break RSA encryption,

and even then we may just use a different method of protecting data.

So there may not be much practical use in searching for Mersenne primes, aside from

winning that $150,000 for running a program while you watch youtube.

I gotta download that software when I get home.

Before you run off and download GIMPS, go ahead and subscribe.

And for more math, find out how ham sandwiches are helping us understand the universe, here.

The largest prime found by hand is M127, so we're obviously a bit past that by now.

Until next time, I'm Julian, don't forget to share that sweet prize money with me when

you win.