Number Theory

1. Jan 22, 2008

k3N70n

Okay I hope it's okay if I have a couple question. I've been strugelling a bit with this problem set. About a quarter of the questions I just don't seem to see how to start them. Any hints would be greatly appreciated. Thank you kindly
I
1. The problem statement, all variables and given/known data

Give that $p\nmid n$ for all primes $p\leq \sqrt[3]n$ show that n> is either prime or the product of two primes.

2. Relevant equations

?

3. The attempt at a solution

I don't really see how to start this one. Any hint would be greatly appreciated

II.
1. The problem statement, all variables and given/known data

Give another proof of the infinitude of primes by assuming that there are only finitely many primes say $p_1, p_2, ... p_n$, and using the following integer to arrive at a a contradiciton:
N = $$p_2p_3...p_n + p_1p_3...p_n +...+p_1p_2...p_{n-1}$$

2. Relevant equations

3. The attempt at a solution

I think that this proof should involve showing that $p_k\nmid N\forall k$ so N must be prime. Which would be like like Euler proof, but I can't seem to see how to set that up

2. Jan 22, 2008

Dick

For I), if all of the primes less than n^(1/3) don't divide n then n still has a prime factorization. How many factors can there be? For II) for any i, p_i divides all of the terms which sum to N except one. Can it divide N?