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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.