## 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
$65537+3+5+17+257=65819$