The Largest Known Prime

A *Mersenne prime* is a prime of the form
.
According to [#!caldwell:largestprime!#] the largest known prime
as of March 2007 is the 44th Mersenne prime

which has 9,808,358 decimal digits. The Electronic Frontier Foundation has offered a $100,000 prize to the first person who finds a 10,000,000 digit prime.

Euclid's theorem implies that there definitely are infinitely many primes bigger than . Deciding whether or not a number is prime is interesting, as a theoretical problem, and as a problem with applications to cryptography, as we will see in Section 2.4 and Chapter 3.

sage: p = 2^32582657 - 1 # this is easy sage: s = p.str(10) # this takes a long time (about a minute) sage: len(s) # s is a very long string (long time) 9808358 sage: s[:20] # the first 20 digits of p (long time) '12457502601536945540' sage: s[-20:] # the last 20 digits (long time) '11752880154053967871'

William 2007-06-01