Number Theory (find the smallest prime divisors)
I am only stuck on part c), but I will show you parts a) and b) to see where this question is coming from.
a) Prove that if p and q are odd primes and q | a^p - 1, then either q | (a - 1) or q = 2kp + 1 for some integer k.
b) Use part a) to show that if p is an odd prime, then the prime divisors of 2^p - 1 are of the 2kp + 1.
c) Find the smallest prime divisors of the integers 2^17 - 1 and 2^29 - 1.
I proved parts a) and b) and tried applying these to part c). For 2^17 - 1 I let a = 2 and p = 17 in parts a) and b). Thus, the prime divisors of 2^17 - 1 are of the form 34k + 1. How do I find the smallest prime of this form that divides 2^17 - 1. If I let k = 3, then I get 103, which is prime, but 103 does not divide 2^17 - 1.
The back of the book has answers for these but I do not know how to get there:
2^17 - 1 is prime
233 | (2^29 - 1)