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Search for the Largest Prime


Date: 08/01/2000 at 02:55:15
From: Jeffrey Kochanski
Subject: Large numbers

As a math teacher, I am often asked what the largest number is. Of 
course, there is no largest number, but an interesting question is, 
"what is the largest finite number that has a practical use in some 
branch of mathematics or science?" How or where has this number been 
used? What notation is used to express this number, if scientific 
notation is not sufficient?

I know that different mathematicians might answer this question 
differently, but that makes it a good question to start a discussion 
in class. What is your opinion on this large number question?

Thank you for your help.
Jeffrey Kochanski


Date: 08/01/2000 at 08:38:26
From: Doctor Paul
Subject: Re: Large numbers

You're right about different people having different opinions. Here's 
what I think:

Since there is no largest integer, it doesn't make much sense to talk 
about that. But what about the largest known prime number?

It turns out that some of the largest knows primes are a special kind 
of prime number called Mersenne Primes (named after the French 
mathematician Marin Mersenne).

A Mersenne Prime is a prime number of the form (2^n)-1. Note that all 
numbers of this form are not prime. For example, (2^4)-1 = 16-1 = 15 
is not prime. But (2^n)-1 is prime for n = 2,3,5 and many other values 
of n.  

The largest known prime is also a Mersenne prime: (2^6972593)-1

There is a great Web site about Mersenne primes at:

   http://mersenne.org/   

One of the things that this Web site does is the "Great Internet 
Mersenne Prime Search" (GIMPS). Let me explain:

One of the recent developments in modern mathematics is called 
parallel (or distributed) computing. This refers to the ability to do 
the same (lengthy) computation on numerous computers and then combine 
the results. So a calculation that might take a year to do on one 
computer can be done in several weeks if the job is split between 
twenty of thirty super computers.

GIMPS harnesses the power of the Internet to have literally millions 
of people checking to find the next prime number. Their software is 
free and runs as a process on the background of your computer. You'll 
never even know it's there.

You should note that (2^n)-1 gets very large very quickly. For 
example, n = 10 yields 1023. You could spend a large amount of time 
trying to do n = 100. An obvious question should be, "How do I tell if 
the number I have chosen is prime?" You can search the Web site 
yourself for an answer to this question.

I think your class could find this very interesting. It deals with a 
relatively new branch of mathematics (made available by the invention 
of the personal computer) and they can even do it themselves at home. 
I even think they're giving away $100,000 to the person who finds the 
first 10-million digit prime number.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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