## The Largest Known Prime

Though Theorem 1.2.1 implies that there are infinitely many primes, it still makes sense to ask the question What is 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 Example 1.2   We can compute the decimal expansion of in SAGE, though watch out as this is a serious computation that may take around a minute on your computer. Also, do not print out or below, because both would take a very long time to scroll by.
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