The exact wording of the question:
"11 and 101 are prime numbers.
a) Show that 1001 is not a prime number.
b) Find the next smallest prime number of the form 100...01 and show that it is the next"
I highly doubt it's intended to infringe on the subject of Generalised Fermat Numbers, so...
"11 and 101 are primes, whereas 1001 is not (7*11*13). Find the next smallest prime of the form 100...01, and show that it is the next"
I'm fairly sure that there no primes of the form 10^k + 1 above 101, but I can't seem to find a complete proof.
Consider the general case of factorising...