You assume there exists a solution and show that it would imply a contradiction.
What you are asking is basically for a method to prove that a generalized fermat number (in this case $\displaystyle 10^{2^n}+1$ ) is not prime over a certain value of $\displaystyle n$ which is an open problem in mathematics as far as I know.
If this homework is from a computer-science class maybe your prof wants you to devise a method to test for primality, in which case Pepin's test would probably be the best approach (for deterministic primality, for probabilistic primality other algorithms may work better.)
Please refer to GFN10 factoring status if you don't believe me that this is an open problem. So far, up to $\displaystyle n=18$ Keller has compiled a list of known prime factors of the generalized fermat numbers, meaning they are all not prime.
You can read more about fermat numbers here
Generalized Fermat Number -- from Wolfram MathWorld
and here
Fermat number - Wikipedia, the free encyclopedia .
As gmatt said, this is an open problem in mathematics. Asking us to solve it for you is like asking us to prove or disprove the Goldbach conjecture or the Riemann hypothesis for you.
It may interest you to know that this exact question has been asked before on another thread in another forum (which I found through a Google search). Over there, someone came up with the spin of interpreting the numbers as given in base 2, for which the question can be easily answered. Except for that spin, the same conclusion was reached over there as over here.