TITLE: A simple method of factorization $ 2^{30}+1 $
QUESTION [5 upvotes]: How can you factor $ 2^{30} + 1 $?  This task was supposed to be at one interview, there is an assumption that there should be a simple solution.

REPLY [11 votes]: Observe that $$x^k+1=\frac{x^{2k}-1}{x^k-1}=\frac{\prod_{d\mid 2k}\Phi_{d}(x)}{\prod_{d\mid k}\Phi_{d}(x)}=\prod_{d\mid 2k,\ d\nmid k}\Phi_{d}(x)$$ With $\Phi_n(x)$ being the $n^\text{th}$ cyclotomic polynomial. It follows that $$2^{30}+1=\Phi_4(2)\Phi_{12}(2)\Phi_{20}(2)\Phi_{60}(2)$$
With the individual factors computed as $$\Phi_4(2)=2^2+1=5$$ $$\Phi_{12}(2)=2^4-2^2+1=13$$ $$\Phi_{20}(2)=2^8-2^6+2^2-2^2+1=5\times 41$$ $$\Phi_{60}(2)=2^{16}+2^{14}-2^{10}-2^8-2^6+2^2+1=61\times 1321$$
Giving an overall factorization of $\boxed{2^{30}+1=5^2\times 13\times 41\times 61\times 1321}$