# Help with 2 Number Theory Problems

SomeRandomGuy
Hey guys, I have a homework assignment for number theory and two of the problems I don't know how do solve. I was hoping I could get some hints or help. Thanks

Problem 1
Let m,n be elements of the natural numbers where n > m.
and F_n = 2^2^n + 1

a.) Show that (F_n - 2)/F_m is an integer.
b.) Assume that there is a prime p in the factorization of both F_n,F_m. Show this leads to a contradiction.
c.) Now there must be atleast n distinct primes. Now let n -> infinity. Write out the proof in detail.

Problem 2
Let p be a prime and let n be any integer satisfying 1 <= n <= p-1. Prove that p divides the binomial coefficient p!/(p-n)!n!

For this, I said if p divides it, then p*a = it where a is some integer. I then get a term that reduces down to a = (p-1)!/(p-n)!n! and don't know where to do from here.

I'm not looking for solutions, although if it's needed for me to understand that's fine. I am just hoping to get pointed in the right direction. Thanks.

Homework Helper
1.a) I would suggest solving by induction on n-m = i. Proving it for the base case (i = 1) is a simple matter of factoring a difference of squares. I assume that the inductive step should not be too hard to prove.
b) If there is a common prime p in the factorizations of Fn and Fm, then if we let Fn = xp and Fm = yp, then from part a we get:

(xp - 2)/(yp) = k for some integer k