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 $2^q-1$ . According to [#!caldwell:largestprime!#] the largest known prime as of March 2007 is the 44th Mersenne prime

$$p = 2^{32582657}-1,$$

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 $p$ . 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 $p$ 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 $p$ or $s$ 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'