Wednesday, February 20, 2013

The Largest Known Prime Number

Quick!  How many prime numbers can you recite, starting at 2 (1 is not prime!)?  Let's see... 2, 3, 5, 7, 11, 13, 17, 19, 23... the further you go, the more you have to think about it, right?  What you have to consider is whether each number you evaluate is only divisible by itself and 1, or if it can be divided by other numbers.  If it's only 1 and itself, you've found a prime number!  But as you count up higher, there is a higher likelihood that these bigger numbers are going to have some out-of-the-ordinary factors that you don't automatically think about.  So then, you can see the role that computers play in finding prime numbers.

Computers can do all of the hard calculations.  They can check to see for any given number if another number can completely divide into it.  This may seem simple if you consider a number like 35 (which isn't prime, because there is 1x35 and also 5x7), but what about considering a number like 64,092,283? It's not likely that you are going to check to see if it has any factors in a short amount of time, but a computer can do it relatively quickly.  Of course, the larger the number, the more calculations it has to check and so even they will take some time.

When you get up to the REALLY large numbers, you have to consider that the computer needs lots of time to evaluate all possible factors for each number.  Since prime numbers become more rare as you count higher, a larger portion of the values that the computer has to spend time evaluating will actually work out to be non-primes.  For this reason, the most recent prime number was discovered FOUR YEARS after the previous prime was reported.

There is currently a distributing computing project named GIMPS (Great Internet Mersenne Prime Search) that is actively doing calculations and checks to find the next prime number.  Curtis Cooper from the University of Central Missouri was the one who reportedly found this latest prime, and it is easy to see why it has taken so long to find.  This largest prime number ever found has more than 17 MILLION digits.  It is the number represented by 2 57,885,161 -1.  Now, imagine having to check that number by hand to see if it had any other factors to rule it out as a prime.  This may not be the most glamorous job for a computer to do, certainly not as entertaining or visually appealing as editing digital photographs or playing video games, but it definitely is a job meant for a computer.

On that note, Phillip Bump got creative with this 17 million digit monster.  He divided it into 6 digit chunks, each chunk then representing an RGB colour, and then plotted out the resulting colour distribution.  From that point of view, the following picture is essentially the fingerprint of this prime.  Maybe working with primes IS a visual art!

The Largest Known Prime Number - by Phillip Bump


  1. Cool picture, but your first sentence has a problem... 1 is NOT prime. If it work the Fundamental Theorem of Arithmetic is faulty as there is not a unique factorization of any number.

  2. Thanks for catching my error, Danny! I've updated the post, and I'm planning on writing another post about primes in the near future where I will talk about the Fundamental Theorem of Arithmetic.


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