Monday Math Musing 2 – New Largest Prime Number – So What?

First, check out this news article from Time Magzine.  Give it a skim.  Then come back.

What happened?

A new largest prime number has been discovered through a program run by a mathematician at the University of Central Missouri.  In case you’ve forgotton, a prime number is one whose factors are merely 1 and itself.  The numbers 3, 5, 7, 11, and 13 are all prime, but 6 isn’t (it can factor into 2*3) and 15 isn’t (3*5).  Prime numbers are really the bread-and-butter of many mathematicians, especially those who study a branch of mathematics called number theory.

Prime numbers have interested mathematicians since ancient times and for a long time mathematicians quested to find the largest prime number.  Unfortunately for them, it is easily proven that there is no largest prime number, or more to the point, there are infinitely many prime numbers.

This of course hasn’t deterred mathematicians from seeking larger and larger of such numbers, and developing methods to make this search easier.  One method was developed by Marin Mersenne, a French theologian, mathematician, and musician from the turn of the 17th century.  He discovered that many prime numbers can actually be written in the form 2^p-1, where p is some other prime number.  For example, the prime number 7 can be written as 2^3-1 (2^3 = 8 – 1 = 7).   Primes of this form are called Mersenne Primes.

Unfortunately, not all numbers of this form are Mersenne primes.  Take for instance 2^11 − 1 = 2,047 = 23*89.  More unfortunately, there doesn’t seem to be a rule for the primes you can use for p that will make a Mersenne prime.  So the only way to tell is to check, which can take a long, long time.

Enter the Great Internet Mersenne Prime Search (or GIMPS).  This group has quested to find increasingly larger and larger Mersenne Primes by using raw computing power to check mind-bogglingly large numbers.  The cool thing about it is you can help.  There is software available at the website that you can download to your computer that will allow you to join into the hunt.  The software will run in the background, using your computer’s idle time to crunch calculations, checking to see if increasingly larger numbers are prime. In fact, all of the largest prime numbers discovered in recent years have been Mersenne primes discovered by users of this program

But you have your work cut out for you.  The prime number discovered in January has p=57,885,161.  The number is more than 17 million digits long (17,425,170 to be exact), which means if you tried to write it down at about 2 digits per second without stopping to eat or sleep, it would take you more than 3 months to write it all and would be about 100 miles long.

Why do we care?

So what good is this largest prime number?  To be honest, it’s worthless.  Having the largest prime number is about as useful has having the largest ball of twine (or even less so because in the latter case at least you have twine).  Searching for the largest prime number is just an exercise in computing power and ingenuity.  Of course, the Electronic Frontier Foundation (EFF) had set aside a $100,000 award for the first prime number with more than 10 million digits, found in 2008, and has reportedly offered a $150,000 prize for the first prime with over 100 million digits, so there’s a nifty financial incentive for the people involved.

That said, prime numbers in general are actually extremely important.  Prime numbers are the basis of public-key cryptography, which is used in just about every Internet transaction from buying things on Amazon, to paying bills with your online bank account, to sending e-mails.  Here’s how it works

I want to send an e-mail to my grandfather.  I have two keys, my private key and my public key.  My private key, which only I myself know, is a pair of really big prime numbers.  My public key, which I send out any time I request a website to do something, is the product of those really big prime numbers.  My grandfather also has two keys, different from mine.

When I send my e-mail, I first ask my grandfather for his public key, then send my e-mail with the rule that the only way to read it is to know what the key factors into.  When my grandfather gets it, he’s able to read it because he knows what the key factors into, it’s his private key.

I’ll explain it another way.  I want to send my grandfather an important package, so he sends me an open padlock that only he has the key for.  I attach the lock to my package and send it back to him, and he uses his key to open it.  At no point does anybody see the key that opens the lock.

This may not seem like a big deal.  After all, the only thing you need to read my e-mail is to know what my grandfather’s public key factors into.  Say his public key is 77.  He plugs 7 and 11 into the lock and it opens, letting him read his e-mail.  But everybody knows that 77 = 7*11.

Look back at what I said.  We’re talking about really big prime numbers here.  I mean really big.  So big that their product is hundreds of digits long.  The point is, there are so many prime numbers that could factor into that large number, that checking them all would take trillions of years to do, longer than the entire universe has even existed!!! Hackers have better things to do (like hacking your system to try to figure out what your private key is).

Of course, someday, some mathematician could figure out a way to do that really quickly and easily.  Should that happen, we’re in a heap of trouble.

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