#math trivia #257 solution

 

Answer:

  • n = 3 gives 2^(2^n) + 1 = 2^(2^3) + 1 = 2^8 + 1 = 257.
  • Other day-numbers of this form are

3 = 2^(2^0) + 1, for n = 0

5 = 2^(2^1) + 1, for n = 1

17 = 2^(2^2) + 1, for n = 2

These are all Fermat numbers, named for the mathematician who studied numbers of this form.  They’re all also Fermat primes, as they both have the form and are prime numbers.

It was once conjectured that all Fermat numbers are primes, but at this point the only Fermat numbers known to be prime are the four mentioned here and the Fermat number for n = 4, which is 65537.  The larger ones that have been studied are all composite — but because they grow so quickly as a function of n, it will take a long time with current methods to determine the primality of even larger ones, and nothing has yet been proved about whether other primes do or don’t exist.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s