Thirty-two Power of Two

Find the sum of the prime factors of 2^{32}-1.
Source: mathcontest.olemiss.edu 6/23/2008

SOLUTION
2^{32}-1=\left (2^{16}\right )^2-1^2
=\left (2^{16}+1\right )\left (2^{16}-1\right)
=\left (65537\right )\left (65535\right )

65537 is the largest known Fermat prime.
65535=3\times 5\times 17\times 257

Therefore,
2^{32}-1=65537\times 3\times 5\times 17\times 257

The sum of the prime factors is
65537+3+5+17+257=65819

Answer: 65819.

Advertisements

About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , . Bookmark the permalink.

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