Is $2^{57} + 1$ is a composite number?

Question

Prove or disprove the following: $2^{57} + 1$ is a composite number.

Answer 1

We can factor $$2^{57}+1 = \left( 2^{19} \right)^3+1 = \left( \left(2^{19} \right)^2 -2^{19} + 1 \right)\left( 2^{19} + 1 \right)$$ where both factors are greater than $1$. Therefore, $2^{57}+1$ is composite.

Answer 2

Yes: $2^{57}+1\equiv(-1)^{57}+1=-1+1=0\mod 3$.