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© 1999-2010 by Francis Edward Su
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From the Fun Fact files, here is a Fun Fact at the Easy level:

Largest Known Primes

Since there are infinitely many primes, what are the largest primes that we know of?

The largest known primes are ones of the form (2m - 1). The reason is that there exist efficient ways to test whether such numbers are prime. Primes of this type are called a Mersenne primes.

As of Sept 2010, the largest known primes were
243,112,609 - 1
242,643,801 - 1
237,156,667 - 1
The largest is over 2 million digits long! These primes were all discovered in the last 3 years; the search for large primes has accelerated with the help of several hundred people across the internet in a project called GIMPS [the Great Internet Mersenne Prime Search]. For more on this, see the URL in the reference.

Presentation Suggestions:
Ask students to guess how large those numbers are, before you tell them.

The Math Behind the Fact:
As it turns out, knowing large primes is very important in cryptography. Being able to factor large numbers is "equivalent" to being able to crack codes, and typical codes that are nearly impossible to break are ones which depend on knowing a large number that is almost prime.

How to Cite this Page:
Su, Francis E., et al. "Largest Known Primes." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:
Keywords:    number theory
Subjects:    number theory
Level:    Easy
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