Show that if $10^{n}+1$ is prime then $n$ must be a power of 2 . Now assuming $n$ is a power of 2 , show that if $p$ is a prime factor of $10^{n}+1$ then $p \equiv 1(\bmod 2 n)$.

Explain the method of Fermat factorization, and use it to factor $10^{4}+1$.