<s>[INST] Find prime factorization of $2^{22} + 1$ Find the prime factorization of $2^{22} + 1$.
How can I approach this with a subtle way?
I know that $a^n -b^n = (a-b) (a^{n-1}+a^{n-2}b + \cdots + ab^{n-2}+b^{n-1})$ and $a^{2n+1} + b^{2n+1} = (a+b) (a^{2n}-a^{2n-1}b + \cdots - ab^{2n-1}+b^{2n})$. So far I have tried to apply the latter one on $4^{11} + 1$. 
 [/INST] $2^{22}+1=2^2\cdot2^{20}+1=4\cdot\Big(2^5\Big)^4+1^4.\quad$ But $4x^4+1=\big(2x^2-2x+1\big)\big(2x^2+2x+1\big)$.
</s>