10. The function $\operatorname{SPF}(n)$ denotes the sum of the prime factors of $n$, where the prime factors are not necessarily distinct. For example, $120=2^{3} \times 3 \times 5$, so $\operatorname{SPF}(120)=2+2+2+3+5=14$. Find the value of $\operatorname{SPF}\left(2^{22}-4\right)$.
Solution
100
Write $2^{22}-4$ as a product of primes:
$$
\begin{aligned}
2^{22}-4 & =4 \times\left(2^{20}-1\right) \\
& =2^{2} \times\left(2^{10}-1\right)\left(2^{10}+1\right) \\
& =2^{2} \times\left(2^{5}-1\right)\left(2^{5}+1\right)(1024+1) \\
& =2^{2} \times(32-1)(32+1)(1025) \\
& =2^{2} \times 31 \times 33 \times 5 \times 205 \\
& =2^{2} \times 31 \times 3 \times 11 \times 5 \times 5 \times 41 \\
& =2^{2} \times 3 \times 5^{2} \times 11 \times 31 \times 41 .
\end{aligned}
$$

Therefore $S P F\left(2^{22}-4\right)=2+2+3+5+5+11+31+41=100$.