Since there are infinitely many primes, what are the largest
primes that we know of?
The largest known primes are ones of the form
(2^{m}  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
2^{43,112,609}  1
2^{42,643,801}  1
2^{37,156,667}  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>.
